AP Calculus AB : Chain rule and implicit differentiation

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #21 : Chain Rule And Implicit Differentiation

Differentiate 

Possible Answers:

Correct answer:

Explanation:

In this chain rule derivative, we need to remember what the derivative of the base function ().  The exponential function is a special function that always returns the original function.  In the chain rule application, we just need to multiply that by the derivative of the exponent.

.

Example Question #21 : Chain Rule And Implicit Differentiation

Find the derivative of .

Possible Answers:

Correct answer:

Explanation:

Be careful with this derivative because it is a hidden chain rule! Let's start with rewriting the problem:

Now, it is easier to notice that it is a chain rule.  A chain rule involves differentiating the outermost function and then, multiplying by the derivative of the inner part.

 

Example Question #201 : Derivatives

Compute the derivative of 

Possible Answers:

Correct answer:

Explanation:

This is a multiple chain derivative.  With chain derivatives, we always want to start on the outermost function (keeping the rest the same), and then, multiply by the inner function derivative until we have take the derivative of every part. 

Example Question #24 : Chain Rule And Implicit Differentiation

Find the derivative: 

 

Possible Answers:

 

Correct answer:

 

Explanation:

 

Implicit differentiation is useful when we have an equation containing a dependent variable , and and an independent variable , but we cannot solve to write  explicitly in terms of 

Differentiate both sides simultaneously: 

 

     

The chain rule must be applied to the following derivatives: 

                       

                  

Noting the  is implied to be a function of , we have to first differentiate with respect to  and then differentiate  with respect to  Since we do not have an explicit definition for , we simply express its' derivative in the equation as , treating it as an "unknown," which needs to be solved for algebraically at a later step.  

For (1):

 

For (2):

 

Using these derivatives the main equation becomes: 

 

Now we simply solve for  and simplify as much as possible. Isolate all terms with a derivative onto one side of the equation and factor: 

  

 

 

 

Example Question #25 : Chain Rule And Implicit Differentiation

Find the derivative of the function: 

Possible Answers:

Correct answer:

Explanation:

On this problem we have to use chain rule, which is: 

So in this problem we let 

 

and 

.

Since 

 

and 

,

we can conclude that 

Example Question #201 : Derivatives

Find the derivative of:  

Possible Answers:

Correct answer:

Explanation:

On this problem we have to use chain rule, which is: 

So in this problem we let 

 

and 

.

Since 

 

and 

,

we can conclude that 

Example Question #202 : Derivatives

Find the derivative of: 

Possible Answers:

Correct answer:

Explanation:

On this problem we have to use chain rule, which is: 

So in this problem we let 

 

and 

.

Since 

 

and 

,

we can conclude that 

 

and 

Example Question #201 : Derivatives

Find the derivative of: 

Possible Answers:

Correct answer:

Explanation:

On this problem we have to use chain rule, which is: 

So in this problem we let 

 

and 

 

and 

.

Since 

 

and 

 

and 

,

we can conclude that 

Example Question #29 : Chain Rule And Implicit Differentiation

Find the derivative of the function: 

Possible Answers:

Correct answer:

Explanation:

On this problem we have to use chain rule, which is: 

So in this problem we let 

 

and 

.

Since 

 

and 

,

we can conclude that 

 

and 

Example Question #30 : Chain Rule And Implicit Differentiation

Find the derivative of: 

Possible Answers:

 

 

 

 

 

Correct answer:

 

Explanation:

On this problem we have to use chain rule, which is: 

So in this problem we let 

 

and 

.

Since 

 

and 

,

we can conclude that 

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