Difference between two derivative definitions

Question

just been very confused about the definition of these two derivatives the past few days.

$\lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h}$

and

$\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$

When do i use them? and what are the differences between these two "definitions" and how come they're called definitions and not formulas?

Answer 1

The Right hand derivative of a function at $x$ is defined as: $$f'(x)=\lim_{h\to 0^+}\frac{f(x+h)-f(x)}{h}$$ If you had to find the right hand derivative of the function at $a$, simply replace $x$ with $a$.

Similarly the left hand derivative is defined as: $$f'(x)=\lim_{h\to 0^+}\frac{f(x-h)-f(x)}{-h}$$

Answer 2

These formulas are the definition of derivative.

One is the derivative at $a$, $f'(a)$; the other is the derivative at $x$, $f'(x)$.

In essence they are the same.