Simple Rules for Differentiation



Sums, Products, and Quotients

Objectives:

Students will be able to

• Calculate the derivatives of sums and differences.

• Apply the product rule to find derivatives.

• Apply the quotient rule to find derivatives.

We continue looking at ways to calculate derivatives using generalized rules. Here we will use the rules for functions form by sums/differences, products, and quotients.

Sums and Differences

If both f and g are differentiable at x, then the sum [pic] and the difference [pic] are differentiable at x and the derivatives are as follows.

[pic] has a derivative [pic]

[pic] has a derivative [pic]

Product Rule

If both f and g are differentiable at x, then the product [pic] is differentiable at x and the derivative is [pic]

Quotient Rule

If both f and g are differentiable at x and [pic], then the quotient [pic] is differentiable at x and the derivative is [pic]

Example 1:

Use the rules of derivatives to find the derivative of [pic].

Example 2:

Use the rules of derivatives to find the derivative of [pic].

Example 3:

Use the product rule to find the derivative of [pic].

Example 4:

Use the rules of derivatives to find the derivative of [pic].

Example 5:

Use the rules of derivatives to find the derivative of [pic].

Example 6:

Use the rules of derivatives to find the derivative of [pic]

Example 7:

The total profit (in tens of dollars) from selling x self-help books is [pic]. Find the average profit from each sales level.

a. 8 books

b. 15 books

c. x books

d. Find the marginal average profit function.

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