differentiating density function

Question

If $X$ is a real-valued random variable with distribution $F$ and density $f$. We have that the distribution and density depend on the parameter $a$. Then since $f$ is density we have $\int f(x,a)dx=1$. Under what conditions is it true that $\int f_a(x,a) dx=0$? It makes intuitive sense but when I try say with the exponential distribution it doesnt work.