derivative of a function depending on a function and chain rules

Question

Given the functions $g(y,z)$ and $ $ $F(x, g(y,z)),$

how to compute the derivative of the 2 following functions w.r.t $ $ $y$ ?

$F(x, g(y,z))\quad $ and $\quad \int_{0}^{y} F\big(y-\xi, g(\xi,z)\big)d\xi.$

Thank you!