Can I apply Fundamental Theorem of Calculus to $\frac{d}{d a}\int_a^{\infty}(x-a)f(x)dx$

Question

I want to find the derivative:
$$\frac{d}{d a}\int_a^{\infty}(x-a)f(x)dx$$

What I have tried is by starting to rewrite the integral:
$$\int_a^{\infty}(x-a)f(x)dx=\int_{\infty}^{a}af(x)dx-\int_{\infty}^{a}xf(x)dx$$
How can I use Fundamental Theorem of Calculus find an expression for the first derivative $d/da$

I might be able to use the technique discussed here
Is there a fundamental theorem of calculus for improper integrals? but I haven't yet found a proper way?

Answer 1

Just write it as$$a\int_\infty^af(x)\,\mathrm dx-\int_\infty^axf(x)\,\mathrm dx.$$Now use the fundamental theorem of Calculus for improper integrals and the rule of the derivative of a product of functions.