The circular rod is being twisted about its axis. There is a rotation of each cross section by an angle ##\theta## that varies linearly with distance along the rod axis. If you work out the shear strain between adjacent cross sections, you obtain ##r\frac{d\theta}{dx}##. For this shear strain, you can then work out the shear stress. Knowing the shear stress, you can then integrate the differential torque on each area of cross section about the axis. This will involve the polar moment of inertia. Knowing the actual torque, you can then determine ##d\theta/dx##. This will be sufficient to determine the maximum shear stress at r = R, the location of maximum shear stress.