Chapter 3 The Derivative

3.2 The Derivative Function

Example: Find the derivative with respect to x of Solution: Example

Example: Find the derivative with respect to x of Solution: Example

Differentiability

Geometrically, a function f is differentiable at x 0 if the graph of f has a tangent line at x 0. The following figures illustrates two common ways in which a function that is continuous at x 0 can fail to be differentiable at x 0 Corner points Points of vertical tangency

3.2.3 THEOREM If a function f is differentiable at x 0, then f is continuous at x 0. Note: Functions are not differentiable at corner points and points of vertical tangency. Function are no differentiable at points of discontinuity (by previous theorem). Differentiability and Continuity

If a function f is defined on [a, b], then f’ is not defined at the endpoints of the interval because derivatives are two-sided limits. We define left-hand derivatives by, and right-hand derivatives by. These are called one-sided derivatives. In general, f is differentiable on an interval of of the form [a, b], [a, + ), (-, b], [a, b) or (a, b] if it is differentiable at all points Inside the interval and appropriate one-sided derivative exists at each included endpoint. Derivatives at the endpoints of an interval

The process of finding a derivative is called differentiation. It can be denoted by The value of the derivative at a point x 0 can be expressed as Other derivative notations