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 #11 : Chain Rule And Implicit Differentiation

.  Find .

Possible Answers:

Correct answer:

Explanation:

To take the derivative, you must first take the derivative of the outside function, which is sine.  However, the , or the angle of the function, remains the same until we take its derivative later.  The derivative of sinx is cosx, so you the first part of  will be .  Next, take the derivative of the inside function, .  Its derivative is , so by the chain rule, we multiply the derivatives of the inside and outside functions together to get .

Example Question #191 : Derivatives

.  Using the chain rule for derivatives, find .

Possible Answers:

Correct answer:

Explanation:

By the chain rule, we must first take the derivative of the outside function by bringing the power down front and reducing the power by one.  When we do this, we do not change the function that is in the parentheses, or the inside function.  That means that the first part of  will be .  Next, we must take the derivative of the inside function.  Its derivative is .  The chain rule says we must multiply the derivative of the outside function by the derivative of the inside function, so the final answer is .

Example Question #12 : Chain Rule And Implicit Differentiation

Use implicit differentiation to find  is terms of  and  for,  

 

 

Possible Answers:

Correct answer:

Explanation:

 

To differentiate the equation above, start by applying the derivative operation to both sides, 

 

Both sides will require the product rule to differentiate, 

 

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 Common Mistake 

A common mistake in the previous step would be to conclude that instead of  . The former is not correct; if we were looking for the derivative with respect to , then  would in fact be . But we are not differentiating with respect to , we're looking for the derivative with respect to 

We are assuming that  is a function of , so we must apply the chain rule by differentiating with respect to  and multiplying by the derivative of  with respect to  to obtain 

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Collect terms with a derivative onto one side of the equation, factor out the derivative, and divide out  to solve for the derivative 

 

 

Therefore, 

 

 

 

 

Example Question #12 : Chain Rule And Implicit Differentiation

Differentiate, 

Possible Answers:

Correct answer:

Explanation:

                          (1)

 

An easier way to think about this:

Because  is a function of a function, we must apply the chain rule. This can be confusing at times especially for function like equation (1). The differentiation is easier to follow if you use a substitution for the inner function, 

Let,

                               (2)

So now equation (1) is simply, 

                               (3)

Note that  is a function of . We must apply the chain rule to find  

                            (4)

 

To find the derivatives on the right side of equation (4), differentiate equation (3) with respect to , then Differentiate equation (2) with respect to 

 

                                                          

 

Substitute into equation (4),  

                  (5)

Now use  to write equation (5) in terms of  alone: 

 

Example Question #11 : Chain Rule And Implicit Differentiation

Find  given 

Possible Answers:

Correct answer:

Explanation:

Here we use the chain rule: 

Let 

Then 

And 

Example Question #11 : Chain Rule And Implicit Differentiation

If , calculate 

Possible Answers:

Correct answer:

Explanation:

Using the chain rule, we have

.

Hence, .

Notice that we could have also simplified  first by cancelling the natural log and the exponential function leaving us with just , thereby avoiding the chain rule altogether.

Example Question #11 : Chain Rule And Implicit Differentiation

Use the chain rule to find the derivative of the function 

Possible Answers:

Correct answer:

Explanation:

First, differentiate the outside of the parenthesis, keeping what is inside the same.

You should get  .

Next, differentiate the inside of the parenthesis. 

You should get .

Multiply these two to get the final derivative .

Example Question #18 : Chain Rule And Implicit Differentiation

Find the derivative of .

Possible Answers:

Correct answer:

Explanation:

Use chain rule to solve this. First, take the derivative of what is outside of the parenthesis.

You should get .

Next, take the derivative of what is inside the parenthesis. 

You should get .

Multiplying these two together gives .

Example Question #19 : Chain Rule And Implicit Differentiation

Find the derivative of .

Possible Answers:

Correct answer:

Explanation:

This is a chain rule derivative.  We must first start by taking the derivative of the outermost function.  Here, that is a function raised to the fifth power.  We need to take that derivative (using the the power rule).  Then, we multiply by the derivative of the innermost function:

Example Question #20 : Chain Rule And Implicit Differentiation

Find the derivative of the following function:

.

Possible Answers:

Correct answer:

Explanation:

This is a chain rule derivative.  We must first differentiate the natural log function, leaving the inner function as is. Recall:

Now, we must replace this with our function, and multiply that by the derivative of the inner function:

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