Multi-Variable Chain Rule - AP Calculus BC
Card 0 of 170
Compute
for
,
,
.
Compute for
,
,
.
All we need to do is use the formula for multivariable chain rule.





When we put this all together, we get.

All we need to do is use the formula for multivariable chain rule.
When we put this all together, we get.
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem




The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem






The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem





The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem










The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem








The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem










The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem,










The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem,
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem,








The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem,
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem,








The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem,
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem,









The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem,
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem,








The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem,
Compare your answer with the correct one above
Use the chain rule to find
when
,
,
.
Use the chain rule to find when
,
,
.
The chain rule states
.
Since
and
are both functions of
,
must be found using the chain rule.
In this problem,













The chain rule states .
Since and
are both functions of
,
must be found using the chain rule.
In this problem,
Compare your answer with the correct one above
Evaluate
in terms of
and/or
if
,
,
, and
.
Evaluate in terms of
and/or
if
,
,
, and
.
Expand the equation for chain rule.

Evaluate each partial derivative.
, 
, 
, 
Substitute the terms in the equation.

Substitute
.

The answer is: 
Expand the equation for chain rule.
Evaluate each partial derivative.
,
,
,
Substitute the terms in the equation.
Substitute .
The answer is:
Compare your answer with the correct one above
Find
where
,
, 
Find where
,
,
To find the derivative of the function with respect to t, we must use the multivariable chain rule, which states that
, where 
Using this rule for both variables, we find that
,
,
, 
Taking the products according to the formula above, and remembering to rewrite x and y in terms of t, we get

To find the derivative of the function with respect to t, we must use the multivariable chain rule, which states that
, where
Using this rule for both variables, we find that
,
,
,
Taking the products according to the formula above, and remembering to rewrite x and y in terms of t, we get
Compare your answer with the correct one above
Find
, where
and
, 
Find , where
and
,
To find the derivative of the function with respect to t, we must use the multivariable chain rule, which states that
for x. (We do the same for the rest of the variables, and add the products together.)
Using the above rule for both variables, we get
,
,
, 
Plugging all of this into the above formula, and remembering to rewrite x and y in terms of t, we get

To find the derivative of the function with respect to t, we must use the multivariable chain rule, which states that for x. (We do the same for the rest of the variables, and add the products together.)
Using the above rule for both variables, we get
,
,
,
Plugging all of this into the above formula, and remembering to rewrite x and y in terms of t, we get
Compare your answer with the correct one above
Find
, where 
Find , where
To find the derivative of the function, we must use the multivariable chain rule. For x, this states that
. (We do the same for y and add the results for the total derivative.)
So, our derivatives are:
,
,
, 
Now, using the above formula and remembering to rewrite x and y in terms of t, we get

To find the derivative of the function, we must use the multivariable chain rule. For x, this states that . (We do the same for y and add the results for the total derivative.)
So, our derivatives are:
,
,
,
Now, using the above formula and remembering to rewrite x and y in terms of t, we get
Compare your answer with the correct one above
Determine
, where
,
,
, 
Determine , where
,
,
,
To find the derivative of the function with respect to t, we must use the multivariable chain rule, which states that, for x,
. (The same rule applies for the other two variables, and we add the results of the three variables to get the total derivative.)
So, the derivatives are
,
,
,
,
, 
Using the above rule, we get

which rewritten in terms of t, and simplified, becomes

To find the derivative of the function with respect to t, we must use the multivariable chain rule, which states that, for x, . (The same rule applies for the other two variables, and we add the results of the three variables to get the total derivative.)
So, the derivatives are
,
,
,
,
,
Using the above rule, we get
which rewritten in terms of t, and simplified, becomes
Compare your answer with the correct one above
Find
, where
,
, 
Find , where
,
,
To find the given derivative of the function, we must use the multivariable chain rule, which states that

So, we find all of these derivatives:
,
,
, 
Plugging this into the formula above, and rewriting x and y in terms of t, we get

which simplified becomes

To find the given derivative of the function, we must use the multivariable chain rule, which states that
So, we find all of these derivatives:
,
,
,
Plugging this into the formula above, and rewriting x and y in terms of t, we get
which simplified becomes
Compare your answer with the correct one above
Find
of the following function:
, where
,
, 
Find of the following function:
, where
,
,
To find the derivative of the function
with respect to t we must use the multivariable chain rule, which states that

Our partial derivatives are:
,
,
,
,
, 
Plugging all of this in - and rewriting any remaining variables in terms of t - we get

To find the derivative of the function with respect to t we must use the multivariable chain rule, which states that
Our partial derivatives are:
,
,
,
,
,
Plugging all of this in - and rewriting any remaining variables in terms of t - we get
Compare your answer with the correct one above