Question

Define differentiability of a function at a point.

Answer 1

Let $f\left(x\right)$ be a real valued function defined on an open interval $(a,b)$ and let $c\in (a,b)$.
Then $f\left(x\right)$ is said to be differentiable or derivable at $x=c$ iff
$\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$ exists finitely.

or, $f\text{'}\left(c\right)=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}.$