1. 3.3: Rules of Differentiation Objective: Students will be able to… Apply the Power Rule, Sum and Difference Rule, Quotient and Product Rule for differentiation Find higher order derivatives

2. What is the slope of y=2? y=-5? y=10?  What is f’(x) if f(x)=2?  DERIVATIVE OF A CONSTANT: PROOF Let f(x) =c, where c is a constant If c is a constant, then

3. EXPLORATION Using the definition of the derivative, find the derivative of each function below: 1.) f(x) = x 1 2.) f(x)=x 2 3.) f(x)=x 3 Do you notice anything about f(x) and f’(x)?

4. The Power Rule: If n is a rational number, then the function f(x)=x n is differentiable and Examples: Find the derivative a.) f(x) = x 4 b.) g(x)= c.) y=

5. The Constant Multiple Rule If f is a differentiable function and c is a real number, then cf is also a differentiable function and Examples:

6. The Sum and Difference Rules If f and g are differentiable functions, then their sum (f +g) and their difference (f-g) are also differentiable and

7. EXAMPLES:

8. Does y=3x 4 -x 2 +1 have any horizontal tangent lines?

9. THE PRODUCT RULE!!! YEAH!!! If f and g are differentiable functions, then their product f·g is differentiable and

10. EXAMPLES

11. THE QUOTIENT RULE!! WOOHOO! If f and g are differentiable functions, then their quotient f/g, g≠0, is differentiable and

12. EXAMPLES:

13. Rewriting Before Differentiating

15. Find the equation of the line tangent to the curve at point (1,-1).

16. Second and Higher Order Derivatives: y’ = dy/dx is the first derivative with respect to x. Second Derivative: Third Derivative Fourth Derivative: Fifth Derivative:

17. Examples: 1.Find y’’ given y=3x 2 +5, 2.Find y (5) given y=x 4 +2x 3 -3x 2 +10x+1.