1. CHAPTER 4 DIFFERENTIATION NHAA/IMK/UNIMAP

2. INTRODUCTION Differentiation – Process of finding the derivative of a function. Notation NHAA/IMK/UNIMAP

3. DERIVATIVE OF A POWER FUNCTION If n is an integer, then: NHAA/IMK/UNIMAP

4. DERIVATIVE OF A CONSTANT If f is differentiable at function x and c is any real number, then c is differentiable: NHAA/IMK/UNIMAP

5. Example 1 Differentiate the following function: NHAA/IMK/UNIMAP

6. DERIVATIVE OF SUM AND DIFFERENCE RULES If f and g are differentiable at function x, then the function f+g and f-g are differentiable: NHAA/IMK/UNIMAP

7. Example 2 Differentiate the following function: NHAA/IMK/UNIMAP

8. Derivative of Trigonometric Functions NHAA/IMK/UNIMAP

9. DERIVATIVE OF EXPONENTIAL & LOGARITHMIC FUNCTIONS NHAA/IMK/UNIMAP

10. PRODUCT RULE If u and v are differentiable at function x, then so the product u.v, thus NHAA/IMK/UNIMAP

11. Example 3: Differentiate the following function: NHAA/IMK/UNIMAP

12. QUOTIENT RULE If u and v are differentiable at function x, then is also differentiable NHAA/IMK/UNIMAP

13. Example 4 Differentiate the following function: NHAA/IMK/UNIMAP

14. Example 5 Differentiate and SIMPLIFY the following function: NHAA/IMK/UNIMAP

15. Example 6 Differentiate the following function: NHAA/IMK/UNIMAP

16. COMPOSITE FUNCTION The Chain Rule – If g is differentiable at point x and f is differentiable at the point g(x), then is differentiable at x. – Let and, then NHAA/IMK/UNIMAP

17. Example 7 Differentiate the following function: NHAA/IMK/UNIMAP

18. “Outside-Inside” Rule – Alternative method for Chain Rule: – If,then NHAA/IMK/UNIMAP COMPOSITE FUNCTION

19. Example 8 Differentiate the following function: NHAA/IMK/UNIMAP

20. These equation define an implicit relation between variables x and y. When we cannot put an equation F(x,y)=0 in the form y = f(x), use implicit differentiation to find NHAA/IMK/UNIMAP IMPLICIT DIFFERENTIATION

21. Differentiate both sides of the equation with respect to x, treating y as a differentiable function of x Collect the terms with on one side of the equation Solve for NHAA/IMK/UNIMAP IMPLICIT DIFFERENTIATION