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 #61 : Chain Rule And Implicit Differentiation

Given the function , find its derivative.

Possible Answers:

Correct answer:

Explanation:

Given the function , we can find its derivative using the chain rule, which states that 

where  and   for . We have  and , which gives us 

Example Question #62 : Chain Rule And Implicit Differentiation

Given the function , find its derivative.

Possible Answers:

Correct answer:

Explanation:

Given the function , we can find its derivative using the chain rule, which states that 

where  and   for . We have  and , which gives us 

Example Question #63 : Chain Rule And Implicit Differentiation

Given the function , find its derivative.

Possible Answers:

Correct answer:

Explanation:

Given the function , we can find its derivative using the chain rule, which states that 

where  and   for . We have  and , which gives us 

Example Question #64 : Chain Rule And Implicit Differentiation

Given the function , find its derivative.

Possible Answers:

Correct answer:

Explanation:

Given the function , we can find its derivative using the chain rule, which states that 

where  and   for . We have  and , which gives us 

Example Question #65 : Chain Rule And Implicit Differentiation

Find .

Possible Answers:

Correct answer:

Explanation:

Let  

Then

can be rewritten as 

Let 

The function can now be rewritten as

Applying the chain rule twice:

Example Question #62 : Chain Rule And Implicit Differentiation

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

To find the derivative of the function, you must apply the chain rule, which is as follows:

Using the function from the problem statement, we have that

 and 

Following the rule, we get

Example Question #65 : Chain Rule And Implicit Differentiation

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

To find the derivative of the function, you must apply the chain rule, which is as follows:

Using the function from the problem statement, we have that

 and 

Following the rule, we get

Example Question #68 : Chain Rule And Implicit Differentiation

Find .

Possible Answers:

None of the other choices gives the correct response.

Correct answer:

Explanation:

Let 

Then 

and

Apply the chain rule:

Substitute back for :

Apply the sum rule:

After some simple algebra:

Example Question #69 : Chain Rule And Implicit Differentiation

 is a function of . Solve for  in this differential equation:

Possible Answers:

Correct answer:

Explanation:

The expressions with  can be separated from those with  by multiplying both sides by :

Find the indefinite integral of both sides:

Set . Then , or , and

Substitute back:

;

Raise  to both powers:

.

 

The correct choice is 

Example Question #70 : Chain Rule And Implicit Differentiation

Find the derivative using the chain rule.

Possible Answers:

Correct answer:

Explanation:

Use the chain rule to find the derivative: 

Thus, 

 

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