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

Polynomials are differentiable for all arguments.

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

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.