Partial derivative of beta function

Theorem: The following formula holds: $$\dfrac{\partial}{\partial x} B(x,y)=B(x,y) \left( \dfrac{\Gamma'(x)}{\Gamma(x)} - \dfrac{\Gamma'(x+y)}{\Gamma(x+y)} \right) = B(x,y)(\psi(x) - \psi(x+y),$$ where $B$ denotes the Beta function, $\Gamma$ denotes the gamma function, and $\psi$ denotes the digamma function.

Proof: █