Use the chain rule to find z(s) and z(t) if f(x,y)=x^2y+2xy^3
x(s,t)=sln(t)
y(s,t)=s^2+t^3
So you want to find $\displaystyle z_s$ and $\displaystyle z_t$ given:
$\displaystyle z=x^2y+2xy^3$
$\displaystyle x(s,t)=s\ln{t}$
$\displaystyle y(s,t)=s^2+t^3 $
Since z depends on both x and y,
$\displaystyle z_s = z_xx_s+z_yy_s$
(treating t as a constant). It's just the chain rule for two variables, except z depends on variables other than s, so you're looking for a partial derivative.
The situation for $\displaystyle z_t$ is similar.
That should get you going in the right direction. Post again if you have any more trouble.
- Hollywood