Define

1. Show that

   $$\begin{array}{rl}& \frac{f(g(x+h))-f(g(x))}{h}\\ =& \frac{f(g(x)+\delta )-f(g(x))}{\delta }\frac{g(x+h)-g(x)}{h}\end{array}$$

2. Explain why

   $$\delta \to 0\text{ as }h\to 0$$

3. Hence, use first principles to prove the chain rule:

   $$\frac{d}{dx}f(g(x))=f^{\prime }(g(x))g^{\prime }(x)$$