I have the following spherical harmonic $$Y_3^1(\upsilon, \varphi) = (-1)^1\sqrt {\frac{(2\times 3 +1)}{4\pi}\frac{(3-1)!}{(3+1)!}}P_3^1(\cos(\upsilon))e^{i\varphi}$$ Which simplifies down to $$-\sqrt{\frac{7}{48\pi}}P_3^1(\cos(\upsilon))e^{i\varphi}$$ Inputting the values for $P_3^1(\cos(\upsilon))$ we obtain the following equation $$P_3^1(\cos(\upsilon)) = \frac{(-1)^1}{2^3\times3!}(1-\cos^2(\upsilon))^\frac{1}{2}\frac{d^{3+1}(\cos^2(\upsilon)-1)}{d\cos^4(\upsilon)}$$ Which simplifies down to $$P_3^1(\cos(\upsilon)) = -\frac{\sin(\upsilon)}{48}\frac{d^4(-\sin^2(\upsilon))^3}{d\cos^4(\upsilon)}$$
This seems unfeasible to solve, does anyone know where I went wrong/improvements that can be made to make it easier?