What is the partial derivative of u with respect to t in terms of x, y, and t?

[jamesbob]

This is annoying me as i have the answer on the tip of my pen, just can't write it down. I'm not 100% sure i understand what the question is asking me to do.

Consider the quantity $$u = e^{-xy}$$ where (x,y) moves in time t along a path:

$$x = \cosh{t}, \mbox{ } y = \sinh{t}$$​

Use a method based on partial derivatives to calculate $$\frac{du}{dt}$$ as a function of x, y and t.

My answer:

I partially differentiated u, getting:

$$\frac{\delta{u}}{\delta{x}} = -ye^{-xy}$$
$$\frac{\delta{u}}{\delta{y}} = -xe^{-xy}$$
So does this mean $$du = -ye^{-xy} + -xe^{-xy} ?$$

I though that i would get du from the part iv just explained, then get dt from differentiating x and y. But this ofcourse leaves me with expressions for dx/dt and dy/dt. Where do i go from here?

 

[neutrino]

Use the chain rule:
$$\frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dx}{dt} + \frac{\partial u}{\partial y} \frac{dy}{dt}$$

 

[jamesbob]

Yeah, realized that after some research - just something id never saw. Dead easy tho. Thanks anyway