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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.

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up vote 4 down vote accepted

In your computation, you forgot the fact that $x$ and $y$ are functions of $t$. Thus, instead of $\frac{dz}{dt} = \frac12 y^2 + 4xy$, you can write: $\frac{dz}{dt} = \frac12 y(t)^2 + 4 x(t) y(t)$, so when $t$ is zero, you get $\frac{dz}{dt} = \frac12 4 + 4 \cdot 5 \cdot 2 = 42$.

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Looks good, but I think the OP made a mistake that would change your answer. I think it should be $\frac12y^2+4xy$ – Mike Nov 6 '12 at 17:34

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