Simple Rules for Differentiation Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and differences. Rules Power Rule a

a 1 f ( x ) x f ( x ) ax

For the function , for all arbitrary constants a. Rules Sums and Differences Rule If both f and g are differentiable at x, then f g f g the sum and the difference are differentiable at x and the derivatives are as follows.

F (x ) f (x ) g(x ) has a derivative F (x ) f (x ) g(x ) Rules Sums and Differences Rule If both f and g are differentiable at x, then f g f g the sum and the difference are differentiable at x and the derivatives are as follows.

G (x ) f (x ) g(x ) has a derivative G (x ) f (x ) g(x ) Example 1 Use the simple rules of derivatives to find the derivative of f (x ) x 6 Example 2

Use the simple rules of derivatives to find the derivative of D( p) 10p 3 2 Example 3 Use the simple rules of derivatives to find the derivative of 6

y 4 x Example 4 Use the simple rules of derivatives to find the derivative of 3 y 6x 15x 2 Example 5 Use the simple rules of derivatives to find the

derivative of 5 p(t ) 12t 6 t t 4 Example 6 Find the slope of the tangent line to the graph of the 3 x = 9. Then find the function y x 4 5x at 2 equation of the tangent line.

Example 7 Find all value(s) of x where the tangent line to the function below is horizontal. 3 2 f (x ) x 5x 6x 3 Example 8 Assume that a demand equation is given by

q 5000 100p Find the marginal revenue for the following levels (values of q). (Hint: Solve the demand equation for p and use the revenue equation R(q) = qp .) a. q = 1000 units b. q = 2500 units c. q = 3000 units Example 9-1 An analyst has found that a companys costs and revenues in dollars for one product are given by the functions

C (x ) 2x and 2 x R (x ) 6x 1000 respectively, where x is the number of items produced. a. Find the marginal cost function. Example 9-2

An analyst has found that a companys costs and revenues in dollars for one product are given by the functions C (x ) 2x and 2 x R (x ) 6x 1000 respectively, where x is the number of items produced.

b. Find the marginal revenue Example 9-3 An analyst has found that a companys costs and revenues in dollars for one product are given by the functions C (x ) 2x and 2 x R (x ) 6x 1000

respectively, where x is the number of items produced. c. Using the fact that profit is the difference between revenue and costs, find the marginal profit function. Example 9-4 An analyst has found that a companys costs and revenues in dollars for one product are given by the functions C (x ) 2x and

2 x R (x ) 6x 1000 respectively, where x is the number of items produced. d. What value of x makes the marginal profit equal 0? Example 9-5 An analyst has found that a companys costs and revenues in dollars for one

product are given by the functions C (x ) 2x and 2 x R (x ) 6x 1000 respectively, where x is the number of items produced. e. Find the profit when the marginal profit is

Example 10-1 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2 M (t ) 3.044t 379.6t 14274.5t 139433 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 Example 10-2 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2 M (t ) 3.044t 379.6t 14274.5t 139433 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. b. 1960 Example 10-3 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2 M (t ) 3.044t 379.6t 14274.5t 139433 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. c. 1980 Example 10-4 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3 2

M (t ) 3.044t 379.6t 14274.5t 139433 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. d. 2000 Example 10-5 The total amount of money in circulation for the years 1915-2002 can be closely approximated by 3

2 M (t ) 3.044t 379.6t 14274.5t 139433 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. e. What do your answers to parts a-d tell you about the amount of money in circulation in those years?