Solution
= Solution
Using the addition law with $y=h$,
$$
\frac{f(x+h)-f(x)}h
=\frac{f(h)}h
\frac{1+f(x)^2}{1-f(x)f(h)}.
$$
Differentiability makes $f$ continuous, so $f(h)\to f(0)=0$, and
$$
\frac{f(h)}h\to f'(0).
$$
Writing $C=f'(0)$ and taking the <limit> gives
$$
\boxed{f'(x)=C(1+f(x)^2)}.
$$