sec8.5.pptx math for business limits and derivatives

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1. Section 5 Basic Differentiation Properties Chapter 8 Limits and the Derivative

2. Brief Review and Derivative Notation Derivative Notation

3. Theorem 1 Constant Function Rule The derivative of any constant function is 0. Example : Suppose f ( x ) = 10π 2 . Find f ′ ( x ). Solution : This function has a constant value, 10π 2 According to Theorem 1, its derivative f ′ ( x ) = 0.

4. Theorem 2 Power Rule Example : Suppose f ( x ) = x 5 . Find f ′ ( x ). Solution : According to Theorem 2, the derivative f ′ ( x ) = 5 x 5-1 = 5 x 4 .

5. Example Differentiating Power Functions

6. Example Differentiating Power Functions

7. Theorem 3 Constant Multiple Property Example : Suppose f ( x ) = 12 x 4 . Find f ′ ( x ). Solution :

8. Theorem 4 Sum and Difference Property Example : Suppose f ( x ) = 3 x 3 + 11 x 2 . Find f ′ ( x ).

9. Example Rewrite before Differentiating

10. Example Equation of Tangent Line Let f ( x ) = 3 x 3 – 2 x 2 + 5. 1.) Find the derivative f ′ ( x ).

11. Example Equation of Tangent Line Let f ( x ) = 3 x 3 – 2 x 2 + 5. 1.) The derivative f ′ ( x ) was found to be f ′ ( x ) = 9 x 2 – 4 x . 2.) Find the equation of the tangent line for x = 1. Solution : 2.)

12. Example Equation of Tangent Line Let f ( x ) = 3 x 3 – 2 x 2 + 5. 1.) The derivative f ′ ( x ) was found to be f ′ ( x ) = 9 x 2 – 4 x . 2.) The tangent line equation was found to be y = 5 x + 1. 3.) Find the values of x where the tangent line is horizontal. Solution : 3.) f ′ ( x ) = 9 x 2 – 4 x