Derivative as a Function

Example

For find the derivative of f and state the domain of f’

( ) 1f x x

The derivative can be regarded as a new function

Example

Given the graph of the function, f , sketch the graph of f’

Example

If find a formula for f’(x). Graph both functions on the calculator and compare.

3( )f x x x

Differentiable

A function is said to be differentiable at a is f’(a) exists. It is differentiable on an open interval (a,b) if it is differentiable at every number on that interval.

If a function is differentiable at a, then it is continuous at a

Some functions can be continuous, but still not differentiabl

Example

Where is differentiable. ( )f x x

Functions that are NOT differentiable

Graphs with kinks or corners do not have tangents at the kinks, so they are not differentiable

Functions that have jump discontinuities at a point are not differentiable at that point

If a tangent line to a function is vertical at a point, the function is not differentiable at that point