CHAPTER 4 DIFFERENTIATION

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

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

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

Example 1 Differentiate the following function:

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:

Example 2 Differentiate the following function:

Derivative of Trigonometric Functions

DERIVATIVE OF EXPONENTIAL & LOGARITHMIC FUNCTIONS

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

Example 3: Differentiate the following function:

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

Example 4 Differentiate the following function:

Example 5 Differentiate and SIMPLIFY the following function:

Example 6 Differentiate the following function:

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

Example 7 Differentiate the following function:

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

Example 8 Differentiate the following function:

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

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