### All AP Calculus AB Resources

## Example Questions

### Example Question #371 : Derivatives

Find the derivative of the function,

**Possible Answers:**

**Correct answer:**

Differentiate both sides and proceed with the product rule:

** (1)**

Evaluate the derivatives in each term. For the first term,

** (2)**

apply the chain rule,

So now the first term in equation (2) can be written,

** (3)**

The second term in equation (2) is easy, this is just the product of multiplied by the derivative of ,

** (4)**

Combine equations (3) and (4) to write the derivative,

### Example Question #2 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative.

**Possible Answers:**

**Correct answer:**

Use the product rule to find the derivative.

### Example Question #3 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative.

**Possible Answers:**

**Correct answer:**

Use the power rule to find the derivative.

Thus, the derivative is

### Example Question #4 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find given

**Possible Answers:**

**Correct answer:**

Here we use the product rule:

Let and

Then (using the chain rule)

and (using the chain rule)

Subbing these values back into our equation gives us

Simplify by combining like-terms

and pulling out a from each term gives our final answer

### Example Question #5 : Derivative Rules For Sums, Products, And Quotients Of Functions

If , evaluate .

**Possible Answers:**

**Correct answer:**

When evaluating the derivative, pay attention to the fact that are constants, (not variables) and are treated as such.

.

and hence

.

### Example Question #1 : Derivative Rules For Sums, Products, And Quotients Of Functions

If , evaluate

**Possible Answers:**

**Correct answer:**

To obtain an expression for , we can take the derivative of using the sum rule.

.

Substituting into this equation gives us

.

### Example Question #2 : Derivative Rules For Sums, Products, And Quotients Of Functions

If , find .

**Possible Answers:**

**Correct answer:**

To find , we will need to use the quotient rule; .

. Start

. Use the quotient rule.

. Take the derivatives inside of the quotient rule. The derivative of uses the product rule.

. Simplify to match the correct answer.

### Example Question #8 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following equation:

**Possible Answers:**

None of the other answers

**Correct answer:**

Because this problem contains two functions multiplied together that can't be simplified any further, it calls for the product rule, which states that .

By using this rule, we get the answer:

By simplifying, we conclude that the derivative is equal to

.

### Example Question #9 : Derivative Rules For Sums, Products, And Quotients Of Functions

**Possible Answers:**

**Correct answer:**

### Example Question #10 : Derivative Rules For Sums, Products, And Quotients Of Functions

**Possible Answers:**

**Correct answer:**