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Chain rule confusion! partial derivatives

  1. Oct 19, 2005 #1
    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!
  2. jcsd
  3. Oct 19, 2005 #2
    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
  4. Oct 19, 2005 #3

    James R

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

    You have:

    [tex]w = xy + yz^2; x = e^t; y = e^t \sin t; z = e^t \cos t[/tex]

    So, you have:

    [tex]w = w(x,y,z); x = x(t); y = y(t); z=z(t)[/tex]

    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:

    [tex]\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}[/tex]
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