The Derivative as a Function

Rather than describing the instantaneous rate-of-change of \(f\) at a specific point \(x=c,\) let’s just keep \(x\) generic and define the derivative as a function of \(x.\) The derivative of \(f\) with respect to \(x\) is the function \(f'\) defined as \[ f'(x) = \lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\,. \] The \(\frac{\mathrm{d}f}{\mathrm{d}x}\) notation alludes to the derivative being a slope; the \(\mathrm{d}\) in that notation may be read as “a small deviation in,” and so \(\frac{\mathrm{d}f}{\mathrm{d}x}\) may be read as “a small deviation in the output per a small deviation in the input.”