]> 6.2 Differentiability of Standard Functions

## 6.2 Differentiability of Standard Functions

All of the standard functions are differentiable except at certain singular points, as follows:

Polynomials are differentiable for all arguments.

A rational function $p ( x ) q ( x )$ is differentiable except where $q ( x ) = 0$ , where the function grows to infinity. This happens in two ways, illustrated by $1 x$ and $1 x 2$ .

Sines and cosines and exponents are differentiable everywhere but tangents and secants are singular at certain values. (Where?)

The inverse functions to powers such as $x 1 / 2$ and $x 1 / 3$ are differentiable where they are defined except where the functions they are inverse to have 0 derivative.