AP Calculus AB : Derivative rules for sums, products, and quotients of functions

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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

Find the derivative of the function, 

 

 

 

Possible Answers:

Correct answer:

Explanation:

 

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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

 

.

and hence

.

Example Question #6 : Derivative Rules For Sums, Products, And Quotients Of Functions

If , evaluate 

Possible Answers:

Correct answer:

Explanation:

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

.

Substituting  into this equation gives us

.

Example Question #7 : Derivative Rules For Sums, Products, And Quotients Of Functions

If , find .

Possible Answers:

Correct answer:

Explanation:

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:

Explanation:

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:

Explanation:

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

Possible Answers:

Correct answer:

Explanation:

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