Proof of Various Derivative Properties — Question 10

Question 10 - Solution

Put $b=f(a)$. Differentiability of $g$ at $b$ means

$$g(b+u)-g(b)=g'(b)u+u\eta(u),$$

where $\eta(u)\to0$ as $u\to0$; define $\eta(0)=0$ so this identity also holds at $u=0$.

For $h\ne0$, set $u=f(a+h)-f(a)$. Then

$$\frac{g(f(a+h))-g(f(a))}{h} =\bigl(g'(b)+\eta(f(a+h)-f(a))\bigr)\frac{f(a+h)-f(a)}h.$$

Differentiability implies continuity of $f$ at $a$, so the first factor tends to $g'(b)$ and the second to $f'(a)$.

$$\boxed{(g\circ f)'(a)=g'(f(a))f'(a).}$$

This argument remains valid when $f(a+h)=f(a)$ for some or all nearby $h$.