Section 2.8 The Derivative as a Function "Life is like music, it must be composed by ear, feeling and instinct, not by rule. Nevertheless one had better know the rules, for they sometimes guide in doubtful cases, though not often." -Samuel Butler

The derivative of a function f at a fixed number a: If we let a vary:

A Visual Exploration <http://hypatia.math.uri.edu/~pakula/DerDraw.htm> Match the Function With Its Derivative <http://www.univie.ac.at/future.media/moe/galerie/diff1/diff1.html>

Example 2 If Find a formula for

Example 3 If , find the derivative of f. State the domain of f’.

Other notations:

Definition: A function f is differentiable at a if f’(a) exists. It is differentiable on an open interval [or or ] if it is differentiable at every number in the interval.

Example Where is the function differentiable?

Theorem: If x is differentiable at a, then f is continuous at a. How can a function fail to be differentiable? A corner A discontinuity A vertical tangent

Higher Derivatives The second derivative:

Example Find and interpret If

The third derivative:

Example Find and interpret If