1. CHAPTER 4
DIFFERENTIATION

2. INTRODUCTION Differentiation
Process of finding the derivative of a function.
Notation

3. DERIVATIVE OF A POWER FUNCTION
If n is an integer, then:

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

5. Example 1 Differentiate the following function:

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:

7. Example 2 Differentiate the following function:

8. Derivative of Trigonometric Functions

9. DERIVATIVE OF EXPONENTIAL & LOGARITHMIC FUNCTIONS

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

11. Example 3: Differentiate the following function:

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

13. Example 4 Differentiate the following function:

14. Example 5 Differentiate and SIMPLIFY the following function:

15. Example 6 Differentiate the following function:

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

17. Example 7 Differentiate the following function:

18. COMPOSITE FUNCTION “Outside-Inside” Rule
Alternative method for Chain Rule:
If ,then

19. Example 8 Differentiate the following function:

20. IMPLICIT DIFFERENTIATION
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

21. IMPLICIT DIFFERENTIATION
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