I'm looking at the Wikipedia page for the derivation of the PDF of the chi-squared distribution, but I'm not sure how they got from the end of the first line to the second line. How do they evaluate the derivative of the integral to end up with what they have?
This is the standard differentiation under the integral sign. In particular, neither the integrand nor the lower limit depends on $y$, hence out of the 3 terms (as seen in wiki) are zero and we have the product between [integrand evaluated at $\sqrt{y}$] and $[\text{upper-limit}]'$.
That is, $$e^{-\frac{t^2}2} \Bigg|_{t = \sqrt{y}} \cdot \frac{d \sqrt{y}}{dy}$$