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

Example Questions

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

Find the derivative of the function,

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.

Explanation:

Use the product rule to find the derivative.

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

Find the derivative.

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

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 .

Explanation:

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

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 .

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

Find the derivative of the following equation:

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

.

Explanation: