Answer
$\phi_x=\frac{a}{\gamma z+ \delta t^2}$, $\phi_y=\frac{2\beta y}{\gamma z+\delta t^2}$, $\phi_z=\frac{-\gamma(\alpha x+\beta y^2)}{(\gamma z+\delta t^2)^2}$, $\phi_t=\frac{-2\delta t(\alpha x+\beta y^2)}{(\gamma z+\delta t^2)^2}$.
Work Step by Step
$\phi(x,y,z,t)=\frac{\alpha x+\beta y^2}{\gamma z+\delta t^2}$
In order to find $\phi_x$ we treat $y$, $z$, and $t$ as constants and differentiate with respect to $x$.
$\phi_x=\frac{a}{\gamma z+ \delta t^2}$
Analogously:
$\phi_y=\frac{2\beta y}{\gamma z+\delta t^2}$
$\phi_z=\frac{-\gamma(\alpha x+\beta y^2)}{(\gamma z+\delta t^2)^2}$
$\phi_t=\frac{-2\delta t(\alpha x+\beta y^2)}{(\gamma z+\delta t^2)^2}$
