Techniques of Differentiation - Quotient Rule

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

Using the first principles, or otherwise, prove that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}\) or equivalently \(\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{u}{v}\right) = \frac{v\frac{\mathrm{d}u}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}\) where \(u = f(x)\) and \(v = g(x)\).

Work on intervals where \(g(x)\ne0\), equivalently \(v\ne0\).