When we see an image through a compound microscope, our image is usually very close to the objective lens. So, in order to get a real image with objective lens (convex in nature), the focus is kept small.

From the above ray diagram,
object distance, $\left(u\right)=d-x$,
image distance, $\left(v\right)=x$,
distance between object and image$=d$,
focal length$=f$.

Using lens formula,

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$$\begin{gathered} \frac{1}{x}-\frac{1}{-\left(d-x\right)}=\frac{1}{f} \\ {x}^2-xd-fd=0 \end{gathered}$$

Real image must be real means roots of the equation should be real.

$$\begin{gathered} \therefore d^2\ge 4fd \\ \therefore d_{\min }=4d \end{gathered}$$