The Derivative from First Principles

The formal definition of the derivative — the one all the shortcut rules are ultimately built from — is the limit of the average rate of change:

provided this limit exists. Finding a derivative directly from this limit is called differentiation from first principles (or ab initio, or by the delta method).

The standard four-step procedure:

1. Write $f(x)$ and form $f(x+h)$ by replacing every $x$ with $x+h$.
2. Compute the difference $f(x+h) - f(x)$ and simplify.
3. Divide by $h$: form $\dfrac{f(x+h)-f(x)}{h}$ and simplify so that the troublesome $h$ in the denominator cancels.
4. Take the limit as $h \to 0$.

Worked illustration — $f(x)=x^2$.

Letting $h \to 0$ gives $f'(x) = 2x$.

The reason step 3 matters: before cancelling, substituting $h=0$ would give the indeterminate form $\tfrac00$. The algebra removes the common factor $h$ so that the limit becomes a legal substitution.