AP Calculus BC : Chain Rule and Implicit Differentiation

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #41 : Computation Of Derivatives

Possible Answers:

Correct answer:

Explanation:

According to the chain rule, . In this case,  and . The derivative is .

Example Question #51 : Computation Of Derivatives

Possible Answers:

Correct answer:

Explanation:

According to the chain rule, . In this case,  and . The derivative is 

Example Question #52 : Computation Of Derivatives

Possible Answers:

Correct answer:

Explanation:

According to the chain rule, . In this case,  and  and 

The derivative is 

Example Question #53 : Computation Of Derivatives

Possible Answers:

Correct answer:

Explanation:

According to the chain rule, . In this case,  and . Since  and , the derivative is 

Example Question #54 : Computation Of Derivatives

Possible Answers:

Correct answer:

Explanation:

According to the chain rule, . In this case,  and . Since  and , the derivative is 

Example Question #55 : Computation Of Derivatives

Possible Answers:

Correct answer:

Explanation:

According to the chain rule, . In this case,  and . Here  and . The derivative is: 

Example Question #56 : Computation Of Derivatives

Given the relation , find .

Possible Answers:

Correct answer:

Explanation:

We begin by taking the derivative of both sides of the equation.

.

. (The left hand side uses the Chain Rule.)

.

.

Example Question #57 : Computation Of Derivatives

Given the relation , find .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

We can use implicit differentiation to find . We being by taking the derivative of both sides of the equation.

 (This line uses the product rule for the derivative of .)

Example Question #58 : Computation Of Derivatives

If , find .

Possible Answers:

Correct answer:

Explanation:

Since we have a function inside of a another function, the chain rule is appropriate here.

The chain rule formula is

.

In our function, both  are 

So we have

and

.

 

Example Question #59 : Computation Of Derivatives

Find  of the following:

Possible Answers:

Correct answer:

Explanation:

To find  we must use implicit differentiation, which is an application of the chain rule.
Taking  of both sides of the equation, we get

using the following rules:

Note that for every derivative with respect to x of a function with y, the additional term dy/dx appears; this is because of the chain rule, where y=g(x), so to speak, for the function it appears in.

Using algebra, we get 

 

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