We are given:
$\displaystyle \frac{1}{f}=\frac{1}{v}+\frac{1}{u}$
Treating $\displaystyle f$ as a constant, can you implicitly differentiate with respect to $\displaystyle u$?
Write this as $\displaystyle f^{-1}= u^{-1}+ v^{-1}$ and differentiate with respect to u.
$\displaystyle 0= -u^{-2}- v^{-2}\frac{dv}{du}$
$\displaystyle v^{-2}\frac{dv}{du}= -u^{-2}$
$\displaystyle \frac{dv}{du}= -u^{-2}v^2$
$\displaystyle \frac{dv}{du}= -\frac{v^2}{u^2}$.