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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