Differentiation of the Beta function

Question

I suppose that \begin{align*} \frac{\partial}{\partial x}\left[B\left(x,y\right)\right]=&\frac{\partial}{\partial x}\left[\int_0^1t^{x-1}(1-t)^{y-1}dt\right]\\ =&\left[\int_0^1\frac{\partial}{\partial x}\left(t^{x-1}(1-t)^{y-1}\right)dt\right]\\ =&\left[\int_0^1(x-1)t^{x-2}(1-t)^{y-1}dt\right]\\ =&(x-1)\left[B(x-1,y)\right] \end{align*} and computing it by Matlab Mupad results in \begin{align*} -\beta(x,y)\left(\Psi(x+y)-\Psi(x) \right) \end{align*} where $\Psi(x)$ is the digamma function. Working with digamma function is not such easy. Can anyone prove that they two are equal or mine is false?

Answer 1

If $x$ is the variable of differentiation, $$\frac{\partial}{\partial x} \left[ t^x \right] \ne x t^{x-1}.$$ That is why you're getting a different result. Instead you should have $$\frac{\partial}{\partial x}\left[ t^x \right] = t^x \log t.$$ And the extra $\log t$ is what introduces the digamma function.