Deriving the Derivative Rules

First Principles, Limit Laws, and the Power Rule

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}

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