Definition of derivative of $f$ at $c$

Question

While solving a proof on "If a function $f$ is differentiable at a point $c$, then it is also continuous at that point". It is stated that

$f$ is differentiable at $c$, we have $$ f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c} $$

Where does it come from ?

My understanding is that $$ f'(c)=\lim_{h\to 0}\frac{f(c+h)-f(c)}{h} $$

Answer 1

With $x=c+h$ we have

$$x \to c \iff h \to 0.$$

Answer 2

If you introduce the variable substitution $x=c+h$, you can see the two limits are identical.