# Derivative of polygamma function

I am working on my Matlab homework and I have to make a derivative of function $$f(x)=\psi (x)\cdot \sin (x)$$ , where $$\psi(x)$$ is polygamma function. What the derivative of $$\psi(x)$$ will be?

• How do we define $\psi(x)$ – George Dewhirst Apr 19 at 19:08

Using $$\psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}$$, the derivative $$\psi'(x) = \frac{\Gamma(x)\Gamma''(x)-\Gamma'(x)^2}{\Gamma(x)^2} = \frac{\Gamma''(x)}{\Gamma(x)}-\psi(x)^2$$,
With $$\Gamma(x)$$ defined as per usual,
$$\Gamma'(x) =$$ "take differential operator inside integral" $$= \int_0^\infty \ln(t)t^{x-1} e^{-t} dt$$
$$\Gamma''(x) = \int_0^\infty \ln(t)^2t^{x-1} e^{-t} dt$$