Differentiation from First Principles

The derivative of \(\mathrm{f}\) at \(x\) is defined by the limiting gradient of a secant as its horizontal increment tends to zero.

First-principles definition

\[\frac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{f}'(x)=\lim_{\delta x\to0}\frac{\mathrm{f}(x+\delta x)-\mathrm{f}(x)}{\delta x}\]

• \(\delta x\) is the change in the input.
• \(\mathrm{f}(x+\delta x)-\mathrm{f}(x)\) is the corresponding change in the output.
• The derivative exists only when this limit exists and is finite.