Chain rule confusion partial derivatives

[mr_coffee]

Hello everyone...
I'm very confused...
i'm suppose to find
dz/dt and dw/dt
but for some of the questions there is no w variable! so what do u put for dw/dt?! Also i have the following:
w = xy + yz^2; x = e^t; y = e^t*sint; z = e^t*cost;
so I'm trying to find dz/dt and dw/dt;
dz/dt = dz/dx * dx/dt + dz/dy * dy/dt + dz/dw * dw/dt;
but when i try dz/dx * dx/dt i need to first take the partial derivative of z with respect to x, but as you can see, z has no x variable! so what do i do about that? THanks!

 

[mathmike]

how do you get dz/dt = dz/dx * dx/dt +dz/dy * dy/dt + dz/dw * dw/dt

dz/dt[e^t * cos t] = cos t * e^t - sint * e^t

there is no dx

as far as dw/dt

it would be

dw/dx * dx/dt + dw/dy * dy/dt + dw/dz + dz/dt

 

[James R]

mr_coffee:

You need to keep track of what is a function of what.

You have:

$$w = xy + yz^2; x = e^t; y = e^t \sin t; z = e^t \cos t$$

So, you have:

$$w = w(x,y,z); x = x(t); y = y(t); z=z(t)$$

If you want to find dz/dt, it's just a simple derivative, since z is only a function of t, and not of x or y or w.

To find dw/dt you need the chain rule:

$$\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt} + \frac{\partial w}{\partial z} \frac{dz}{dt}$$