How to solve combinations of differentials and integrals? In the first chapter of Nearing's book "Mathematical tools for physics" (available online) I encountered an interesting combination of differentials and integrals - which I don't fully understand:
(a) $$\frac{\mathrm{d} }{\mathrm{d}\alpha}\int_{-\infty}^{\infty}e^{-\alpha x^2}dx=-\int_{-\infty}^{\infty}x^2e^{-\alpha x^2}dx$$
(b) $$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{x}e^{-x t^2}dt=e^{-x^{3}}-\int_{0}^{x}t^2e^{-x t^2}dt$$
(c) $$\frac{\mathrm{d} }{\mathrm{d} x}\int_{x^2}^{\,\sin x}e^{x t^2}dt=e^{x\, \sin^2 x} \,\cos x-e^{x^{5}}2x+\int_{x^2}^{\,\sin x}t^2e^{x t^2}dt$$
I can't see in which order you have to do which rules to arrive at the solutions. Could anyone please give me the steps in between? Thank you!

It's a Leibniz integral rule
 -see e.g. in wikipedia by the link.
