## Monday Math 44

Now that we know how to find the derivatives of the gamma function, we can also find the partial derivatives of the beta function. We know that
.
Thus
.
We see from the symmetry between x and y that
.

Finding the second partial derivatives, we see first that
.
Similarly,
.

So we just need the mixed partial derivative


These have a number of applications to definite integrals involving logarithms.