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

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- Oct 15th 2012, 05:42 AMallstar2Need help with a differential equation
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 - Oct 15th 2012, 06:54 PMchiroRe: Need help with a differential equation
Hey allstar2.

Can you show us what you have tried? Do you know what the chain rule says? - Oct 16th 2012, 07:02 AMhollywoodRe: Need help with a differential equation
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