Many students first meet the derivative rules in a first year calculus course. Plenty never see where those rules come from or why functions with certain forms follow certain rules. Here I build the primary rules straight from limits and the limit definition of the derivative. The derivative is the limit definition and everything else follows. We start with a few facts about limits.
Limit Definition of a Derivative
For a function \(f(x)\) the derivative at \(x\) is
\[
f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.
\]
We will use standard limit properties:
\[
\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)
\]
\[
\lim_{x\to a}c\,f(x)=c\cdot\lim_{x\to a}f(x)\quad\text{for constant }c
\]
\[
\lim_{x\to a}(f(x)\,g(x))=\bigl(\lim_{x\to a}f(x)\bigr)\bigl(\lim_{x\to a}g(x)\bigr)
\]
\[
\lim_{x\to a}(f(x))^{n}=\bigl(\lim_{x\to a}f(x)\bigr)^{n}.
\]
Power Rule from the Definition
Let \(f(x)=x^{n}\). Then
\[
f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
=\lim_{h\to 0}\frac{(x+h)^{n}-x^{n}}{h}.
\]
Expand \((x+h)^n\) with the Binomial Theorem
\[
(x+h)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}h^{k}
=x^{n}+\binom{n}{1}x^{n-1}h+\binom{n}{2}x^{n-2}h^{2}+\cdots+h^{n}.
\]
Subtract \(x^n\)
\[
(x+h)^n-x^{n}=\binom{n}{1}x^{n-1}h+\binom{n}{2}x^{n-2}h^{2}+\cdots+h^{n}.
\]
Divide by \(h\)
\[
\frac{(x+h)^n-x^{n}}{h}
=\binom{n}{1}x^{n-1}+\binom{n}{2}x^{n-2}h+\cdots+h^{n-1}.
\]
Take the limit as \(h\to 0\). Every term with a factor of \(h\) vanishes and the first term survives
\[
\lim_{h\to 0}\frac{(x+h)^n-x^{n}}{h}
=\binom{n}{1}x^{n-1}.
\]
Since \(\binom{n}{1}=n\) we get the familiar rule
\[
f'(x)=n\,x^{n-1}.
\]