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 #31 : 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 

Example Question #32 : 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 

 

and 

Example Question #33 : 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 

Example Question #31 : 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 

Example Question #34 : 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 

 

and 

.

Since 

 

and 

 

and 

,

we can conclude that 

Example Question #35 : Chain Rule And Implicit Differentiation

Find dy/dx 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 #37 : Chain Rule And Implicit Differentiation

Find the derivative of the following equation:

Possible Answers:

None of the other answers

Correct answer:

Explanation:

Because we are differentiating a function within another function, we must use the chain rule, which gives that 

.

Using this rule, we see that

,

and therefore, the differentiation of 

 

is 

.

 

Example Question #31 : Chain Rule And Implicit Differentiation

Find the derivative of the following equation:

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Because we are differentiating a function within another function, we must use the chain rule, which states that 

Chain rule.

Looking at our function, we should be able to tell that 

 

and 

.

Given this, we can use the chain rule to solve:

.

Example Question #36 : Chain Rule And Implicit Differentiation

Find the derivative of the following function:

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Because we are differentiating a function within another function, we must use the chain rule, which states that 

Chain rule.

By examining the given equation 

,

we see that we can find the derivative by pulling out the 5, as it is simply a constant:

.

We can see from this that 

 

and 

.

By plugging this information into the chain rule, we find that the derivative is 

.

Example Question #40 : Chain Rule And Implicit Differentiation

Find the derivative of the following equation:

Possible Answers:

Correct answer:

Explanation:

Because we are differentiating a function within another function, we must use the chain rule, which states that 

Chain rule.

from the given equation, 

,

we can deduce that in this case, 

 and .

By plugging this into the chain rule, we find that 

.

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