Derivative Part 2

Definition The derivative of a function f is another function f ’ (read “f prime”) whose value at any number x is : Provided that this limit exists and is not  or -   If this limit does exist  f differentiable at c  Other way  if f differentiable at x1 then f ‘(x1) exist  If a function differentiable at every riil number in their domain then f called differentiable function

Soo if x1 belong to domain then

Add Note : If we take then

Differentiability Implies Continiuty Ex. Check if continue at x=0 and differentiable at x=0?

The Constant Rule

The Power Rule

The Constant Multiple Rule

The Sum and Difference Rules

Derivatives of Sine and Cosine Functions

The Product Rule

The Quotient Rule

Derivatives of Trigonometric Function

Leibniz Notation for Derivatives Ultimately, this notation is a better and more effective notation for working with derivatives.

Teorema If and differentiable function then

The Chain Rule

The General Power Rule

Summary of Differentiation Rules

Exercise 1 Suppose f with Find a and b such as f continue at x=0 but f’(0) does’nt exist

Exercise 2 Check if the function differentiable at 0 ??

Ex3 Check if the function Differentiable at x=0

Ex 4 Find the derivative from the function :

Ex 5 Calculate d/dx(  x  ) then show the function y=  x  satisfied yy’=x, x  0

Ex 6 Find the derivative from the invers function