Derivatives - AP Calculus AB
Card 0 of 7953
Find 
if the function 
is given by
Find
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To find the derivative at 
, we first take the derivative of 
. By the derivative rule for logarithms,
Plugging in 
, we get
To find the derivative at
Plugging in
Find the derivative of the following function at the point 
.
Find the derivative of the following function at the point
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Use the power rule on each term of the polynomial to get the derivative,
Now we plug in 
Use the power rule on each term of the polynomial to get the derivative,
Now we plug in
Let 
. What is 
?
Let
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We need to find the first derivative of f(x). This will require us to apply both the Product and Chain Rules. When we apply the Product Rule, we obtain:
In order to find the derivative of 
, we will need to employ the Chain Rule.
We can factor out a 2x to make this a little nicer to look at.
Now we must evaluate the derivative when x = 
.
The answer is 
.
We need to find the first derivative of f(x). This will require us to apply both the Product and Chain Rules. When we apply the Product Rule, we obtain:
In order to find the derivative of
We can factor out a 2x to make this a little nicer to look at.
Now we must evaluate the derivative when x =
The answer is
Let 
.
Find the second derivative of 
.
Let
Find the second derivative of
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The second derivative is just the derivative of the first derivative. So first we find the first derivative of 
. Remember the derivative of 
is 
, and the derivative for 
is 
.
Then to get the second derivative, we just derive this function again. So
The second derivative is just the derivative of the first derivative. So first we find the first derivative of
Then to get the second derivative, we just derive this function again. So
Define 
.
What is 
?
Define
What is
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Take the derivative 
of 
, then take the derivative of 
.
Take the derivative
Define 
.
What is 
?
Define
What is
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Take the derivative 
of 
, then take the derivative of 
.
Take the derivative
Define 
.
What is 
?
Define
What is
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Rewrite:
&space;=&space;e%5E%7B-4x%7D "f(x) = $e^{-4x}$")
Take the derivative 
of 
, then take the derivative of 
.
Rewrite:
Take the derivative
Define 
.
What is 
?
Define
What is
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Take the derivative 
of 
, then take the derivative of 
.
Take the derivative
What is the second derivative of 
?
What is the second derivative of
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To get the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
Remember that anything to the zero power is one.
Now we do the same process again, but using 
as our expression:
 "$\frac{\mathrm{d}$ }{\mathrm{d} $x}(2x+5x^0$$)=(1*2x^{1-1}$$)+(0*5x^{0-1}$)")
Notice that 
, as anything times zero will be zero.
Anything to the zero power is one.
To get the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
Remember that anything to the zero power is one.
Now we do the same process again, but using
Notice that
Anything to the zero power is one.
What is the second derivative of 
?
What is the second derivative of
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To get the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat 
as 
, as anything to the zero power is one.
That means this problem will look like this:
Notice that 
as anything times zero will be zero.
Remember, anything to the zero power is one.
Now to get the second derivative we repeat those steps, but instead of using 
, we use 
.
Notice that 
as anything times zero will be zero.
To get the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat
That means this problem will look like this:
Notice that
Remember, anything to the zero power is one.
Now to get the second derivative we repeat those steps, but instead of using
Notice that
What is the second derivative of 
?
What is the second derivative of
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To get the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat 
as 
, as anything to the zero power is one.
Notice that 
, as anything times zero is zero.
Now we repeat the process using 
as the expression.
Just like before, we're going to treat 
as 
.
To get the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat
Notice that
Now we repeat the process using
Just like before, we're going to treat
If 
, what is 
?
If
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The question is asking us for the second derivative of the equation. First, we need to find the first derivative.
To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat 
as 
since anything to the zero power is one.
Notice that 
since anything times zero is zero.
Now we do the exact same process but using 
as our expression.
As stated earlier, anything to the zero power is one.
The question is asking us for the second derivative of the equation. First, we need to find the first derivative.
To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat
Notice that
Now we do the exact same process but using
As stated earlier, anything to the zero power is one.
What is the second derivative of 
?
What is the second derivative of
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To find the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat 
as 
since anything to the zero power is one.
 "$\frac{\mathrm{d}$ }{\mathrm{d} $x}(x+1)=(1*x^{1-1}$$)+(0*x^{0-1}$)")
Notice that 
since anything times zero is zero.
That leaves us with 
.
Simplify.
As stated earlier, anything to the zero power is one, leaving us with:
Now we can repeat the process using 
or 
as our equation.
As pointed out before, anything times zero is zero, meaning that 
.
To find the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat
Notice that
That leaves us with
Simplify.
As stated earlier, anything to the zero power is one, leaving us with:
Now we can repeat the process using
As pointed out before, anything times zero is zero, meaning that
What is the second derivative of 
?
What is the second derivative of
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To find the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat 
as 
since anything to the zero power is one.
Notice that 
since anything times zero is zero.
Just like it was mentioned earlier, anything to the zero power is one.
Now we repeat the process using 
as our expression.
Like before, anything times zero is zero.
Anything to the zero power is one.
To find the second derivative, first we need to find the first derivative.
To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat
Notice that
Just like it was mentioned earlier, anything to the zero power is one.
Now we repeat the process using
Like before, anything times zero is zero.
Anything to the zero power is one.
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.
Define &space;=&space;x%5E%7B3%7D&space;-&space;%5Cfrac%7B21%7D%7B2%7D&space;x%5E%7B2%7D&space;+30x&space;-6 "f \left ( x\right ) = $x^{3}$ - $\frac{21}{2}$ $x^{2}$ +30x -6")
.
Give the interval(s) on which 
is increasing.
Define
Give the interval(s) on which
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is increasing on those intervals at which %3E0 "f'(x)>0")
.
We need to find the values of 
for which 
. To that end, we first solve the equation:
These are the boundary points, so the intervals we need to check are:
, 
, and 
We check each interval by substituting an arbitrary value from each for 
.
Choose 
increases on this interval.
Choose 
decreases on this interval.
Choose 
increases on this interval.
The answer is that 
increases on 
We need to find the values of
These are the boundary points, so the intervals we need to check are:
We check each interval by substituting an arbitrary value from each for
Choose
Choose
Choose
The answer is that
Define 
.
Give the interval(s) on which 
is decreasing.
Define
Give the interval(s) on which
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is decreasing on those intervals at which %3C0 "g'(x)<0")
.
&space;=&space;3%5Ccdot&space;x%5E%7B3-1%7D&space;-2%5Ccdot&space;9x%5E%7B2-1%7D+24+0 "g'(x) = 3\cdot $x^{3-1}$ -2\cdot $9x^{2-1}$+24+0")
We need to find the values of 
for which &space;=&space;3x%5E%7B2%7D&space;-18x+24&space;%3C0 "g'(x) = $3x^{2}$ -18x+24 <0")
. To that end, we first solve the equation:
These are the boundary points, so the intervals we need to check are:
, 
, and 
We check each interval by substituting an arbitrary value from each for 
.
Choose 
increases on this interval.
Choose 
decreases on this interval.
Choose 
increases on this interval.
The answer is that 
decreases on 
.
We need to find the values of
These are the boundary points, so the intervals we need to check are:
We check each interval by substituting an arbitrary value from each for
Choose
Choose
Choose
The answer is that
At what value of 
does 
shift from decreasing to increasing?
At what value of
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To find out when the function shifts from decreasing to increasing, we 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.
From here, we want to know if there is a point at which graph changes from negative to positive. Plug in zero for y and solve for x.
This is the point where the graph shifts from decreasing to increasing.
To find out when the function shifts from decreasing to increasing, we 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.
From here, we want to know if there is a point at which graph changes from negative to positive. Plug in zero for y and solve for x.
This is the point where the graph shifts from decreasing to increasing.