How to differentiate this functions?

For a single valued function it is easy to differentiate $f(x+g(t))$ and $f(xg(t))$ for some independent variable $t$ but in vector valued function this is not quite easy to differtiate that

let $f : \mathbb{R^n} \to \mathbb{R}$ , $g: \mathbb{R }^m \to \mathbb{R}$what is $\frac{\partial h}{\partial t_j}, \frac{\partial h}{\partial x_i} $ for $h(x,t)= f(xg(t))$

let $f : \mathbb{R^n} \to \mathbb{R}$ , $g: \mathbb{R }^m \to \mathbb{R}^n$what is $\frac{\partial h}{\partial t_j}, \frac{\partial h}{\partial x_i} $ for $h(x,t)=f(x+g(t))$

I tried to use the chain rule but I didn't get anything