Math Help - Leibnez Rule

1. Leibnez Rule

Hi all, I have to solve the following problem:

Find dV/dt, where:

$\int_0^{t^2} (x+at)dF(x)$ where F(x) is the cdf of x on [0, $\infty$) and f(x) is its corresponding pdf.

My question is can I just use Leibnez rule as one normally would, or does the whole cdf thing complicate matters? My senses tell me its the former.

2. If "F(x) is the cdf of x on [0,) and f(x) is its corresponding pdf" then $dF(x)= f(x)dx$ so the integral is
$\int_0^t^2(x+ at)f(x)dx$ and its derivative, by the Leibniz rule, is simply $2t(t^2+ at)f(t^2)$.