Chain rule

By definition,

$$\frac{\mathrm{d}}{\mathrm{d}x} f(g(x)) = \lim_{h \to 0} \frac{f(g(x+h)) - f(g(x))}{h}$$

Upon substituting $a = x + h$, This can be written as

$$\begin{eqnarray*} \lim_{x \to a} \frac{f(g(x)) - f(g(a))}{x - a} & = & \lim_{x \... ...lim_{x \to a} \frac{g(x) - g(a)}{x - a} \\ & = & f'(g(x)) g'(x) \end{eqnarray*}$$

(N.B. This is only a sketch proof.) This can also be written

$$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}t} \frac{\mathrm{d}t}{\mathrm{d}x}$$

This result is used to differentiate functions of functions, such as . In this case, we put , so that . Hence, .