For the Leibniz integral rule where we have
$\begin{aligned} \frac{d}{d\alpha}\int_{a(\alpha)}^{b(\alpha)} f(x,\alpha)\,dx &= \frac{d b(\alpha)}{d \alpha}\,f(b(\alpha),\alpha)-\frac{d a(\alpha)}{d \alpha}\,f(a(\alpha),\alpha)\\ +& \int_{a(\alpha)}^{b(\alpha)}\frac{\partial}{\partial \alpha}\,f(x,\alpha)\,dx\end{aligned} $
I'm trying to 'apply' this to all different types of integral, for example, I can see that if $a$ and $b$ are just constants then the first 2 terms after the integral sign disappear. Also, if $f$ is just a function of $x$ and not $\alpha$ then this just becomes the equation in the FTC.
But what happens when we differentiate with respect to $x$ instead, so, $ \frac{d}{dx}\int_{a(\alpha)}^{b(\alpha)} f(x,\alpha)\,dx \, ?$
Or perhaps if $a$ and $b$ are functions of $x$? So, $ \frac{d}{d\alpha}\int_{a(x)}^{b(x)} f(x,\alpha)\,dx \, ?$ I could go on...
Are these completely different 'rules' with different equations? Or do they have trivial results that I'm just missing? Do they even make sense in mathematics?
Thanks for any answers!