The question is:
Suppose that the equations $rz^3+sz=t$ and $\sin(s+t)=rs$ determine $z$ as a function of $r,s,t$ and $s$ as a function of $r$ and $t$. Hence, $z$ can be treated as a function $z=f(r,t)$. Find $\frac{\partial f}{\partial t}$.
The answer key writes:
$$\frac{\partial f}{\partial t} = \frac{\partial z}{\partial s}\frac{\partial s}{\partial t} + \frac{\partial z}{\partial t}$$
I dont understand why the chain rule is as such.
$z = z(s,r,t)$, and $s = s(r,t)$, so $$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial s}\frac{\partial s}{\partial t} + \frac{\partial z}{\partial r}\frac{\partial r}{\partial t} + \frac{\partial z}{\partial t}\frac{\partial t}{\partial t}\\ = \frac{\partial z}{\partial s}\frac{\partial s}{\partial t} + \frac{\partial z}{\partial r}0 + \frac{\partial z}{\partial t}1$$