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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.

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