real analysis - How to Prove something is differentiable

I know how differentiability is defined in terms of continuity of the function $(f(x) - f(x_0)) / (x - x_0) $ as $x$ goes to $x_0$, but I was wondering if there are other useful theorems / lemmas I could use to show a function is differentiable?

Note: I am aware of the technique that if I can express my function in terms of a sum/product/quotient of functions that I know are differentiable, then I can just use the product rule, etc. to find the derivatives on top of showing that the function is differentiable.

But are there other lemmas or theorems that are also helpful? (For example, an equivalent definition of continuity is that preimages of open sets are open)

Answer

There are several theorems that you did not mention:

• if $f$ and $g$ are differentiable, then $g\circ f$ is differentiable too (and $(g\circ f)'=(g'\circ f)\times f'$);


• if $f$ is invertible and $f'$ is never $0$, then $f^{-1}$ is differentiable too (and $(f^{-1})'=\frac1{f'\circ f^{-1}}$);


• if $(f_n)_{n\in\mathbb N}$ is a sequence of differentiable functions wich converges pointwise to a function $f$ and if the sequence $(f_n')_{n\in\mathbb N}$ converges uniformly to a function $g$, then $f$ is differentiable (and $f'=g$);


• if $f$ is continuous and $F(x)=\int_a^xf(t)\,\mathrm dt$, then $f$ is differentiable (and $F'=f$).