Differentiation

It all starts from this one simple formula:

A function of a real variable $f(x)$ is differentiable at a point $a$ of its domain, if its domain contains an open interval containing $a$, and the limit

This is really the only formula we need for all of differentiation and here’s how you want to understand it:

Think of $h$ as a small step away from $a$. The quantity $\frac{f(a+h) - f(a)}{h}$ is the slope of the secant line connecting the points $(a, f(a))$ and $(a+h, f(a+h))$. As $h$ gets closer and closer to $0$, the secant line approaches the tangent line to the curve at $x = a$, and its slope becomes the derivative $f’(a)$, i.e., how does the function change with a small change $h$.