Answer
$$z_x = \frac{1}{x + t^2}.$$
$$z_t = \frac{2t}{x + t^2}.$$
Work Step by Step
To find $z_x$, we treat $t$ as a constant and differentiate with respect to $x$. Thus
$$z_x = \frac{\frac{\delta}{\delta x}(x + t^2)}{x + t^2} = \frac{1}{x + t^2}.$$
Similarly, to find $z_t$, we treat $x$ as constant and differentiate with respect to $t$. Thus
$$z_t = \frac{\frac{\delta}{\delta t}(x + t^2)}{x + t^2} = \frac{2t}{x + t^2}.$$
