OPERATIONS WITH DERIVATIVES. The derivative of a constant times a function Justification.

OPERATIONS WITH DERIVATIVES

The derivative of a constant times a function

Justification

Example 1

Example 2

EXERCISE 1

Calculate the derivative of each of the following functions. Then find the equation of the tangent line at the indicated point.

Finding the critical points of a function y=f(x) defined by a formula

• STEP 1. Find all the points where f ‘(x)=0 or where f ‘ (x) is undefined. These are the candidates to be critical points

• STEP 2. From the candidate points choose the ones in the domain of the function f(x). Those are the critical points

Example 2

Find the critical points of the function

• Formula for y’(x)• Solve

– y’(x)=0– y’(x) undefined

• Choose the solutions which are in the domain of y(x)

Exercise 2

Algebraically, find the critical points for each of the following functions

Additive Property of Derivatives

Example 3

Main Operation: Addition

Main Operation: Addition

Derivative of the product of two functions

First Function

Second Function

Exercise 4

Each of the following functions can be represented as the product of functions f and g. Identify the functions and use the the product rule to find its derivative.

Derivative of the quotient of two functions

EXERCISE 5

Each of the following functions can be represented as the quotient of two functions f

and g. Clearly identify them and use the quotient rule to find its derivative. Some of

these derivatives can be calculated using previous rules as well. When it is possible you

are asked to do it.

Applications

• Work on the applications• Work on the practice on computation of

derivatives.