Differential of definite integral

Question

What is the solution of $$ \frac{d}{dy} \int_{0}^{y} \exp(-y(x+1))\, dx $$

If there were no $y$ inside the exponential function the answer would be the exponential function.

Answer 1

$$...=\frac{d}{dy}\exp(-y(y+1))+\int_0^y\frac{\partial }{\partial y}\exp(-y(x+1))\mathrm d x.$$

In general, $$\frac{d}{dt}\int_{g(t)}^{h(t)}f(x,k(t))\mathrm dx=\frac{d}{dt}f(g(t),k(t))-\frac{d}{dt}f(h(t),k(t))+k'(t)\int_{g(t)}^{h(t)}\frac{\partial }{\partial y}f(x,k(t))\mathrm d x.$$