Derivative by Method of First Principle

Finding a derivative straight from its defining limit — without invoking any differentiation rule established later — is called finding the derivative by method of first principle (also called 'ab initio' or 'from the definition'). For convenience, the increment $\delta x$ is renamed $h$.

If $f(x)$ is a function defined on an open interval, its derivative with respect to $x$ by method of first principle is

and the derivative of $y=f(x)$ at the specific point $x=a$, by the same method, is

$f'(a)=\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}=\left(\dfrac{dy}{dx}\right)_{x=a}$ …