Consider a thin spherical shell of radius $r$ and thickness $dr$

Current flows radially between the two concentric spheres, so the area available for current at radius $r$ is the surface area of a sphere: $A = 4\pi r^2$

Write the differential resistance

For a thin shell, $dR = \rho \frac{dr}{A} = \rho \frac{dr}{4\pi r^2}$

Integrate from inner sphere $r=a$ to outer sphere $r=b$

$R = \int_a^b dR = \int_a^b \rho \frac{dr}{4\pi r^2}$

$R = \frac{\rho}{4\pi} \int_a^b \frac{dr}{r^2}$

Evaluate the integral

$\int \frac{dr}{r^2} = \int r^{-2}dr = -\frac{1}{r}$

Therefore, $R = \frac{\rho}{4\pi} \left[-\frac{1}{r}\right]_a^b$

$R = \frac{\rho}{4\pi} \left(-\frac{1}{b} + \frac{1}{a}\right)$

$R = \frac{\rho}{4\pi} \left(\frac{1}{a} - \frac{1}{b}\right)$

Match with the options

This corresponds to: $\boxed{\frac{\rho}{4\pi}\left(\frac{1}{a}-\frac{1}{b}\right)}$

So the correct option is D.