Simple Rules for Differentiation
Simple Rules for Differentiation
Objectives:
Students will be able to
• Apply the power rule to find derivatives.
• Calculate the derivatives of sums and differences.
As you may have noticed, calculating derivatives using the Newton’s Quotient and limits process can be time consuming. Luckily, the process produces that same type of result for certain functions. In this section, we will generalize these results for power functions and functions formed by adding/subtracting power functions to form rules that we will be able to use in place of the Newton’s Quotient and limits process.
Power Rule
For the function [pic], [pic] for all arbitrary constants a.
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]
Example 1:
Use the simple rules of derivatives to find the derivative of [pic].
Example 2:
Use the simple rules of derivatives to find the derivative of [pic].
Example 3:
Use the simple rules of derivatives to find the derivative of [pic].
Example 4:
Use the simple rules of derivatives to find the derivative of [pic].
Example 5:
Use the simple rules of derivatives to find the derivative of [pic].
Example 6:
Find the slope of the tangent line to the graph of the function [pic] at [pic]. Then find the equation of the tangent line.
Example 7:
Find all value(s) of x where the tangent line to the function [pic] is horizontal.
Example 8:
Assume that a demand equation is given by [pic]. Find the marginal revenue for the following levels (values of q). (Hint: Solve the demand equation for p and use the revenue equation [pic].)
a. 1000 units
b. 2500 units
c. 3000 units
Example 9:
An analyst has found that a company’s costs and revenues in dollars for one product are given by the functions [pic] and [pic] respectively, where x is the number of items produced.
a. Find the marginal cost function.
b. Find the marginal revenue function.
c. Using the fact that profit is the difference between revenue and costs, find the marginal profit function.
d. What value of x makes the marginal profit equal 0?
e. Find the profit when the marginal profit is 0.
Example 10:
The total amount of money in circulation for the years 1915-2002 can be closely approximated by [pic] where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.
a. 1920
b. 1960
c. 1980
d. 2000
e. What do your answers to parts a-d tell you about the amount of money in circulation in those years?
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