Techniques of Differentiation - Differentiation by First Principles

Let \(y = f[g(x)]\) be a composite function of functions \(f\) and \(g\) such that \(f\) and \(g\) are differentiable everywhere on their domains, respectively.

Using the first principles, or otherwise, prove that \(\frac{\mathrm{d}y}{\mathrm{d}x} = f'(g(x))g'(x)\), or equivalently \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \times \frac{\mathrm{d}u}{\mathrm{d}x}, \text{ where}\hspace{0.5em} u = g(x).\)