### All AP Calculus AB Resources

## Example Questions

### Example Question #11 : Chain Rule And Implicit Differentiation

. Find .

**Possible Answers:**

**Correct answer:**

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:**

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:**

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:**

** (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:**

Here we use the chain rule:

Let

Then

And

### Example Question #11 : Chain Rule And Implicit Differentiation

If , calculate

**Possible Answers:**

**Correct answer:**

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:**

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:**

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:**

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:**

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: