Derivative as a function - AP Calculus AB
Card 0 of 847
When 
, what is the concavity of the graph of 
?
When
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To find the concavity, we need to look at the first and second derivatives at the given point.
To take the first derivative of this equation, use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent:
Simplify:
Remember that anything to the zero power is equal to one.
The first derivative tells us if the function is increasing or decreasing. Plug in the given point, 
, to see if the result is positive (i.e. increasing) or negative (i.e. decreasing).
Therefore the function is increasing.
To find out if the function is convex, we need to look at the second derivative evaluated at the same point, 
, and check if it is positive or negative.
We're going to treat 
as 
since anything to the zero power is equal to one.
Notice that 
since anything times zero is zero.
Plug in our given value:
Since the second derivative is positive, the function is convex.
Therefore, we are looking at a graph that is both increasing and convex at our given point.
To find the concavity, we need to look at the first and second derivatives at the given point.
To take the first derivative of this equation, use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent:
Simplify:
Remember that anything to the zero power is equal to one.
The first derivative tells us if the function is increasing or decreasing. Plug in the given point,
Therefore the function is increasing.
To find out if the function is convex, we need to look at the second derivative evaluated at the same point,
We're going to treat
Notice that
Plug in our given value:
Since the second derivative is positive, the function is convex.
Therefore, we are looking at a graph that is both increasing and convex at our given point.
At the point 
, is 
increasing or decreasing, and is it concave or convex?
At the point
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To find out if the function is increasing or decreasing, we need to look at the first derivative.
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
Anything to the zero power is one.
Now we plug in our given value and find out if the result is positive or negative. If it is positive, the function is increasing. If it is negative, the function is decreasing.
Therefore, the function is decreasing.
To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave.
To find the second derivative, we repeat the process using 
as our expression.
We're going to treat 
as 
.
Notice that 
since anything times zero is zero.
As stated before, anything to the zero power is one.
Since we get a positive constant, it doesn't matter where we look on the graph, as our second derivative will always be positive. That means that this graph is going to be convex at our given point.
Therefore, the function is decreasing and convex at our given point.
To find out if the function is increasing or decreasing, we need to look at the first derivative.
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
Anything to the zero power is one.
Now we plug in our given value and find out if the result is positive or negative. If it is positive, the function is increasing. If it is negative, the function is decreasing.
Therefore, the function is decreasing.
To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave.
To find the second derivative, we repeat the process using
We're going to treat
Notice that
As stated before, anything to the zero power is one.
Since we get a positive constant, it doesn't matter where we look on the graph, as our second derivative will always be positive. That means that this graph is going to be convex at our given point.
Therefore, the function is decreasing and convex at our given point.
At the point 
, is the function 
increasing or decreasing, concave or convex?
At the point
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First, let's find out if the graph is increasing or decreasing. For that, we need the first derivative.
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
We're going to treat 
as 
since anything to the zero power is one.
Notice that 
since anything times zero is zero.
Plug in our given point for 
. If the result is positive, the function is increasing. If the result is negative, the function is decreasing.
Our result is negative, therefore the function is decreasing.
To find the concavity, look at the second derivative. If the function is positive at our given point, it is concave. If the function is negative, it is convex.
To find the second derivative we repeat the process, but using 
as our expression.
As you can see, our second derivative is a constant. It doesn't matter what point we plug in for 
; our output will always be negative. Therefore our graph will always be convex.
Combine our two pieces of information to see that at the given point, the graph is decreasing and convex.
First, let's find out if the graph is increasing or decreasing. For that, we need the first derivative.
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
We're going to treat
Notice that
Plug in our given point for
Our result is negative, therefore the function is decreasing.
To find the concavity, look at the second derivative. If the function is positive at our given point, it is concave. If the function is negative, it is convex.
To find the second derivative we repeat the process, but using
As you can see, our second derivative is a constant. It doesn't matter what point we plug in for
Combine our two pieces of information to see that at the given point, the graph is decreasing and convex.
A jogger leaves City 
at 
. His subsequent position, in feet, is given by the function:
,
where 
is the time in minutes.
Find the acceleration of the jogger at 
minutes.
A jogger leaves City
where
Find the acceleration of the jogger at
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The accelaration is given by the second derivative of the position function:
For the given position function:
,
,
.
Therefore, the acceleration at 
minutes is 
. Again, note the units must be in 
.
The accelaration is given by the second derivative of the position function:
For the given position function:
Therefore, the acceleration at
Practicing the chain rule level 3 A!
Find the derivative of the function
Practicing the chain rule level 3 A!
Find the derivative of the function
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To understand why the answer is
,
first remember that 
.
Then, you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, 
and 
.
The reason why we did not define 
, 
and 
is because of the property 
mentioned before,
turning
into
in terms of 
, 
and 
is actually 
which means
since 
in 
is substituted with 
and 
in 
can be substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
which is 
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
first remember that
Then, you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
The reason why we did not define
is because of the property
turning
into
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
Practicing the chain rule level 3 B!
Find the derivative of the function
Practicing the chain rule level 3 B!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, 
and 
.
in terms of 
, 
and 
is actually 
which means
since 
in 
is substituted with 
and 
in 
can be substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(z) first, then differentiate it
Step 2: Look at the next function g(z), keep it inside the other function f'(z)
Step 3: Look at the next function h(z), keep it inside the other function g(z)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(z) but multiply it by the factors f'(g(h(z)))*g'(h(z))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(z), g(z), g'(z), h(z) and h'(z) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to differentiate. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(z) first, then differentiate it
Step 2: Look at the next function g(z), keep it inside the other function f'(z)
Step 3: Look at the next function h(z), keep it inside the other function g(z)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(z) but multiply it by the factors f'(g(h(z)))*g'(h(z))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(z), g(z), g'(z), h(z) and h'(z) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to differentiate. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
A gun sends a bullet straight up with a launch velocity of 220 ft/s. It reaches a height of $s=220t-16t^2$ after 
seconds. What's the acceleration in 
of the block after it has been ejected?
A gun sends a bullet straight up with a launch velocity of 220 ft/s. It reaches a height of $s=220t-16t^2$ after
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Since $a=\frac{ds^2$$}{dt^2$}$, by differentiating the position function twice, we see that acceleration is constant and 
. Acceleration, in this case, is gravity, which makes sense that it should be a constant value!
Since $a=\frac{ds^2$$}{dt^2$}$, by differentiating the position function twice, we see that acceleration is constant and
The speed of a car traveling on the highway is given by the following function of time:
Consider a second function:
What can we conclude about this second function?
The speed of a car traveling on the highway is given by the following function of time:
Consider a second function:
What can we conclude about this second function?
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Notice that the function 
is simply the derivative of 
with respect to time. To see this, simply use the power rule on each of the two terms.
Therefore, 
is the rate at which the car's speed changes, a quantity called acceleration.
Notice that the function
Therefore,
The function 
is a continuous, twice-differentiable functuon defined for all real numbers.
If the following are true:
Which function could be 
?
The function
If the following are true:
Which function could be
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To answer this problem we must first interpret our given conditions:
Implies the function is strictly increasing.
Implies the function is strictly concave down.
We note the only function given which fufills both of these conditions is 
.
To answer this problem we must first interpret our given conditions:
We note the only function given which fufills both of these conditions is
Find the eqation of the line tangent to the graph of $y=x+x^{1/2}$ at point 
.
Find the eqation of the line tangent to the graph of $y=x+x^{1/2}$ at point
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$y=x+x^{1/2}$
$y'=1+\frac{1}{2}$x^{-1/2}$
$y'=1+\frac{1}{2(x^{1/2}$$)}
Now we plug in 
.
$y'=1+\frac{1}{2(4^{1/2}$$)} = 1+\frac{1}{4}$ = $\frac{5}{4}$ this is the slope.
Now use the point slope formula to find the tangent line.
y-6=\frac{5}{4}$(x-4)
y-6=\frac{5}{4}$x-5
y=\frac{5}{4}$x+1
$y=x+x^{1/2}$
$y'=1+\frac{1}{2}$x^{-1/2}$
$y'=1+\frac{1}{2(x^{1/2}$$)}
Now we plug in
$y'=1+\frac{1}{2(4^{1/2}$$)} = 1+\frac{1}{4}$ = $\frac{5}{4}$ this is the slope.
Now use the point slope formula to find the tangent line.
y-6=\frac{5}{4}$(x-4)
y-6=\frac{5}{4}$x-5
y=\frac{5}{4}$x+1
Practicing the chain rule level 3 C!
Find the derivative of the function
Practicing the chain rule level 3 C!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
Then, you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, 
and 
.
in terms of , and is actually which means
since in 
is substituted with 
and in 
can be substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
Then, you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
Practicing the chain rule level 1 A!
Find the derivative of the function
Practicing the chain rule level 1 A!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually 
, which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
Practicing the chain rule level 1 B!
Find the derivative of the function
Practicing the chain rule level 1 B!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually , which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
Practicing the chain rule level 1 C!
Find the derivative of the function
Practicing the chain rule level 1 C!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually , which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
Practicing the chain rule level 1 D!
Find the derivative of the function
Practicing the chain rule level 1 D!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of 
and 
is actually 
and
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
Practicing the chain rule level 1 E!
Find the derivative of the function
Practicing the chain rule level 1 E!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, and .
The reason why we did not define , and
is because of the property mentioned before,
turning
into
in terms of , and is actually which means
since in is substituted with and in can be substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
which is
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
The reason why we did not define
is because of the property
turning
into
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
Practicing the chain rule level 2 A!
Find the derivative of the function
Practicing the chain rule level 2 A!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually and
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
Practicing the chain rule level 2 B!
Find the derivative of the function
Practicing the chain rule level 2 B!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of 
and 
is actually 
which means
since 
in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(t), g(t) and g'(t) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(t), g(t) and g'(t) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
Practicing the chain rule level 2 C!
Find the derivative of the function
Practicing the chain rule level 2 C!
Find the derivative of the function
Tap to see back →
To understand why the answer is
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
Practicing the chain rule level 2 D!
Find the derivative of the function
Practicing the chain rule level 2 D!
Find the derivative of the function
Tap to see back →
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations