Prove the equality : Multivariable chain rule problem

[michonamona]

Homework Statement

Prove that

$$(\frac{\partial u}{\partial x})^{2} + (\frac{\partial u}{\partial t})^{2} = e^{-2s}[(\frac{\partial u}{\partial s})^{2} + (\frac{\partial u}{\partial t})^{2}].$$

Homework Equations

$$u = f(x,y)$$
$$x = e^{s}cost$$
$$y = e^{s}sint$$

The Attempt at a Solution

I started out by computing $$\frac{\partial u}{\partial s}$$, then solving it for $$\frac{\partial u}{\partial x}$$. Then I did the same for $$\frac{\partial u}{\partial y}$$. So I got some messy equations, that made think that there must be a much easier way to solve this. I also tried implicit differentiation but got stuck. Any insight?

Thanks,
M

 

[Mark44]

Start by evaluating the right side, and don't try to solve for u_x or u_y. If you calculate the two partials on the right side correctly, many terms will drop out.

 

[michonamona]

Ah...thank you.