In an experiment to measure focal length (f) of convex lens, the least counts of the measuring scales for the position of object (u) and for the position of image (v) are $$\Delta u$$ and $$\Delta v$$, respectively. The error in the measurement of the focal length of the convex lens will be:

The lens formula relates the focal length f to the object distance u and the image distance v as $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$.

Differentiating this expression gives

$$ -\frac{df}{f^2} = -\frac{dv}{v^2} + \frac{du}{u^2} .$$

Considering maximum errors in u and v leads to

$$ \frac{|\Delta f|}{f^2} = \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2} ,$$

which can be rearranged to

$$ \Delta f = f^2\left(\frac{\Delta u}{u^2} + \frac{\Delta v}{v^2}\right) .$$

Therefore, the correct answer is Option 3: $$f^2\left[\frac{\Delta u}{u^2} + \frac{\Delta v}{v^2}\right].$$