When an object is positioned at a distance u from a simple...

The angular magnification $m$ of a simple microscope is defined as the ratio of the angle subtended by the image at the eye ($\theta_i$) to the angle subtended by the object at the unaided eye when placed at the least distance of distinct vision $D$ ($\theta_o$).

Assuming the eye is close to the lens, the angle subtended by the image is $\theta_i = \dfrac{h'}{v}$. Using the linear magnification formula $h' = h$\left(\dfrac{v}{u}\right), we get $\theta_i =$\dfrac{h(v/u)}{v} = \dfrac{h}{u}.

The angle subtended by the object at the unaided eye is $\theta_o =$\dfrac{h}{D}.

Therefore, the angular magnification is $m =$\dfrac{\theta_i}{\theta_o} = \dfrac{h/u}{h/D} = \dfrac{D}{u}.

Thus, for a given object distance $u$, the angular magnification depends only on $u$ (and the constant $D$), and is independent of the focal length $f$.