According to wikipedia, $$\frac{d }{dx}(\int_{a(x)}^{b(x)}f(x,t)dt)=f(x,b(x))\frac{d }{dx}b(x)-f(x,a(x))\frac{d }{dx}a(x)+\int_{a(x)}^{b(x)}\frac{\partial }{\partial x}f(x,t)dt$$
I'm not sure where the last part ($\int_{a(x)}^{b(x)}\frac{\partial }{\partial x}f(x,t)dt$) came from.
Here's my attempt:
$$\frac{d }{dx}(\int_{a(x)}^{b(x)}f(x,t)dt)$$ $$=F(x,b(x))-F(x,a(x))$$ $$=f(x,b(x))\frac{d }{dx}b(x)-f(x,a(x))\frac{d }{dx}a(x)$$
However, the last part is missing. I used AI to help, but it just skipped all the steps even if I wanted it to show them. I also found the result using the first principle, but I think it is also possible to show it without using it. Is it possible to show $\int_{a(x)}^{b(x)}\frac{\partial }{\partial x}f(x,t)dt$ comes out without using the special case a(x)=a and b(x)=b?
