What's equivalent to the following expression?
$\displaystyle \frac { \partial} {\partial p}\int _p^{p_0} f(\alpha,\beta)d\alpha $
Assume that $\displaystyle \int f(\alpha,\beta)\,d\alpha = F(\alpha,\beta)$. Thus,
$\displaystyle \int_p^{p_0}f(\alpha,\beta)\,d\alpha = F(\alpha,\beta)\big|_p^{p_0}=F(p_0,\beta)-F(p,\beta)$
Note that all operations you are doing are either w.r.t. $\displaystyle \alpha$ or $\displaystyle p$, so $\displaystyle \beta$ and $\displaystyle p_0$ should be treated as constants, and therefore $\displaystyle F(p_0,\beta)$ is a constant. Call it $\displaystyle C$.
Now we want $\displaystyle \frac{\partial}{\partial p}[C-F(p,\beta)]=\frac{\partial}{\partial p}[-F(p,\beta)]$ because the derivative of a constant is zero.
Remember how we defined $\displaystyle F$ though. We defined it so that $\displaystyle F~'=f$. So
$\displaystyle \frac{\partial}{\partial p}[-F(p,\beta)]=\boxed{-f(p,\beta)}$