Techniques of Differentiation - Differentiation by First Principles

Let \(y = f(x)g(x)\) such that \(f\) and \(g\) are differentiable everywhere on their domains, respectively.

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