Answer
$h_x=2xy\cos{(\frac{z}{t})}$, $h_y=x^2\cos{(\frac{z}{t})}$, $h_z=\frac{-x^2y\sin{(\frac{z}{t})}}{t}$, $h_t=\frac{x^2yz\sin{(\frac{z}{t})}}{t^2}$.
Work Step by Step
$h(x,y,z,t)=x^2y\cos{(\frac{z}{t})}$
In order to find $h_x$ we treat $y$, $z$, and $t$ as constants and differentiate with respect to $x$.
$h_x=2xy\cos{(\frac{z}{t})}$
Analogously:
$h_y=x^2\cos{(\frac{z}{t})}$
$h_z=\frac{-x^2y\sin{(\frac{z}{t})}}{t}$
$h_t=\frac{x^2yz\sin{(\frac{z}{t})}}{t^2}$
