$A$ short object of length $L$ is placed along the principal axis of a concave mirror away from focus. The object distance is $u$. If the mirror has a focal length $f,$ what will be the length of the image? You may take $L << |u-f|$.

(N/A) The mirror formula is given by $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$.
Differentiating both sides with respect to $u$,we get $-\frac{1}{v^2} \frac{dv}{du} - \frac{1}{u^2} = 0$.
Therefore,$\frac{dv}{du} = -\frac{v^2}{u^2}$.
Since $v = \frac{fu}{u-f}$,we have $\frac{dv}{du} = -\left(\frac{fu}{u-f}\right)^2 \cdot \frac{1}{u^2} = -\left(\frac{f}{u-f}\right)^2$.
The length of the image $L'$ is given by $|dv| = |\frac{dv}{du}| \cdot L$.
Substituting the value of $\frac{dv}{du}$,we get $L' = \left(\frac{f}{u-f}\right)^2 L$.