Let $z=xy^2, dx/dt=\frac{1}{\sqrt{4+t^3}}, dy/dt=e^t\sqrt{4+t}, x(0)=5, y(0)=2$. I want to determine $dz/dt$ when $t=0$.
My computation is that $dz/dx=y^2$ and $dz/dy=2xy$, so therefore $dz/dt=y^2\cdot \frac{1}{\sqrt{4+t^3}} + 2xy\cdot e^t\sqrt{4+t},$ and so subbing in $t=0$ I get $dz/dt=\frac12 y^2 + 2xy$. Does this make sense? I don't think I should be getting $x$ and $y$'s left in here.