Partial derivative of an integral from zero to infinity

Question

How would one go about taking the derivative of this integral? $$\frac{\partial}{\partial C_T} \int_{0}^{\infty} U(C_T)e^{-\delta t}dt$$

Answer 1

$$ \int_{0}^{\infty} U(C_T)e^{-\delta t}dt=U(C_T)\int_{0}^{\infty} e^{-\delta t}dt=\frac{U(C_T)}{\delta} $$ and $$\frac{\partial}{\partial C_T} \int_{0}^{\infty} U(C_T)e^{-\delta t}dt=\frac{U'(C_T)}{\delta}. $$