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Function composition

  1. Dec 12, 2013 #1
    1. The problem statement, all variables and given/known data

    Be z=F(u,v,w), u=f(x,y), v=e-αx, w=ln y, get the expression [itex]\partial[/itex]z/[itex]\partial[/itex]x, [itex]\partial[/itex]z/[itex]\partial[/itex]y.

    2. Relevant equations

    Chain rule.

    3. The attempt at a solution

    [itex]\partial[/itex]z/[itex]\partial[/itex]x=[itex]\partial[/itex]z/[itex]\partial[/itex]v*dv/dx=-α e-αx

    [itex]\partial[/itex]z/[itex]\partial[/itex]y=[itex]\partial[/itex]z/[itex]\partial[/itex]w*dw/dy=1/y
     
  2. jcsd
  3. Dec 12, 2013 #2

    ShayanJ

    User Avatar
    Gold Member

    When calculating [itex] \frac{\partial z}{\partial x} [/itex],you should consider all variables that depend on x not only v!
     
  4. Dec 12, 2013 #3
    Thank you. What about the following expression: ∂z/∂x=∂z/∂u*∂u/∂x+∂z/∂v*dv/dx?. Same to ∂z/∂y with w.
     
  5. Dec 12, 2013 #4

    Mark44

    Staff: Mentor

    Looks good. You even picked up on the fact that dv/dx is a regular (not partial) derivative.
    What did you get?
     
  6. Dec 12, 2013 #5
    ∂z/∂y=∂z/∂u*∂u/∂y+∂z/∂w*dw/dy.
     
  7. Dec 12, 2013 #6

    Mark44

    Staff: Mentor

    Looks good!
     
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