Can I write this integral differently? $$\frac{d}{du}\int_{g(0)}^{g(u)}f(u,t)\, dt$$ I have an integral of a function $f$ which has variables $u$ and $t$. Ordinarily I could just move the $d/du$ inside since the integral is with respect to $t$. But the bound on the integral is a function of $u$. For what it's worth, $g$ is a linear function.

Yes, use the Leibniz Rule:

\begin{align} {d\over du}\int_{g(0)}^{g(u)} f(t,u)\,dt&=\int_{g(0)}^{g(u)} {\partial f\over \partial u}(g(u),u)\,dt+f(g(u),u)g'(u)-f(g(0),u){d\over du}[g(0)]\\\ &=\int_{g(0)}^{g(u)} {\partial f\over \partial u}(g(u),u)\,dt+f(g(u),u)g'(u). \end{align}