How to find the derivative w.r.t. lower limits?

Question

How to find $\frac{d}{dy} \int_{y}^{\infty} \int_{2y}^{\infty} y f(x_2)dx_2 f(x_1)dx_1$, where $x_1$ and $x_2$ are two independent continuous random variables and $f(x_1)$ and $f(x_2)$ are their PDFs. Can I still use Leibniz's rule?

Answer 1

In the general case: let $f$ be an integrable function with antiderivative $F$, then

\begin{align*} \frac{d}{dy}\int_y^a f(x)dx = \frac{d}{dy}\left( F(a)-F(y)\right) = -f(y). \end{align*}

And if we manipulate your expression a bit it becomes \begin{align*} \int_{y}^{\infty} \int_{2y}^{\infty} y f(x_2)dx_2 f(x_1)dx_1 = y\int_{y}^{\infty} f(x_1)dx_1\int_{2y}^{\infty} f(x_2)dx_2 \end{align*} which you can now easily differentiate w.r.t. $y$ using the product rule and what I wrote above.