First-Principles Derivation

First-principles derivation

For two points \((a,f(a))\) and \((a+\delta x,f(a+\delta x))\), the secant gradient is \(m=\frac{f(a+\delta x)-f(a)}{\delta x}\).

As \(\delta x\to0\), the secant approaches the tangent, so \(f'(a)=\lim_{\delta x\to0}\frac{f(a+\delta x)-f(a)}{\delta x}\).

For \(f(x)=x^3\), \(f'(x)=\lim_{\delta x\to0}\frac{(x+\delta x)^3-x^3}{\delta x}\).

Expanding and cancelling gives \(f'(x)=\lim_{\delta x\to0}\left(3x^2+3x\delta x+(\delta x)^2\right)\).

Therefore \(f'(x)=3x^2\), so \(\frac{\mathrm d}{\mathrm dx}(x^3)=3x^2\).