First, let's understand the relationship between the object distance ($$u$$), the image distance ($$v$$), and the focal length ($$f$$) of a convex lens, which is given by the lens formula:

Now, to find the error in the focal length ($$\Delta f$$) due to the errors in the measurements of $$u$$ and $$v$$ ($$\Delta u$$ and $$\Delta v$$, respectively), we have to differentiate the lens formula with respect to $$u$$ and $$v$$, keeping in mind the propagation of error.

By differentiating both sides of the lens formula with respect to $$v$$ and $$u$$, and also considering the negative reciprocal relation (given $$f$$ is a constant for a specific lens), we have:

Rearranging this equation to find $$\Delta f$$, we get:

Therefore, the correct option showing the error in the measurement of the focal length of the convex lens, taking into account the least counts ($$\Delta u$$ and $$\Delta v$$) of the measuring scales for the position of the object ($$u$$) and for the position of the image ($$v$$), is:

Option C: $$f^2\left[\frac{\Delta \mathrm{u}}{\mathrm{u}^2}+\frac{\Delta \mathrm{v}}{\mathrm{v}^2}\right]$$