$\begin{aligned} & y=\tan ^{-1}\left(\frac{5 x+1}{3-x-6 x^2}\right)=\tan ^{-1} \frac{(3 x+2)+(2 x-1)}{1-(3 x+2)(2 x-1)} \\ & \Rightarrow y=\tan ^{-1}(3 x+2)+\tan ^{-1}(2 x-1) \\ & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{1+(3 x+2)} \times 3+\frac{1}{1+(2 x-1)^2} \times 2 \\ & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3}{9 x^2+12 x+5}+\frac{2}{4 x^2-4 x+2} \\ & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3}{9 x^2+12 x+5}+\frac{1}{2 x^2-2 x+1}\end{aligned}$