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Chain rule in partial derivative

  1. Sep 19, 2009 #1
    There is a theorem in partial derivative
    If x= x(t) , y= y(t), z= z(t) are differentiable at [tex]t_{0}[/tex], and if w= f(x,y,z) is differentiable at the point (x,y,z)=(x(t),y(t),z(t)),then w=f(x(t),y(t),z(t)) is differentiable at t and
    [tex]\frac{dw}{dt}[/tex]=[tex]\frac{\partial w}{\partial x}[/tex][tex]\frac{dx}{dt}[/tex] + [tex]\frac{\partial w}{\partial y}[/tex] [tex]\frac{dy}{dt}[/tex] + [tex]\frac{\partial w}{\partial z}[/tex] [tex]\frac{dz}{dt}[/tex]
    where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at (x,y,z)

    My question are
    1. What is the difference between dx and[tex]\partial x[/tex]?
    2. i don't know why i am not allowed to eliminate the dx and [tex]\partial x[/tex],also the same for y and z to get 3[tex]\frac{\partial w}{dt}[/tex]
  2. jcsd
  3. Sep 19, 2009 #2


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    The problem appears to be that you do not understand the derivative itself. Neither "[itex]\partial x[/itex]" nor "dx" have any meaning alone. And, when using the "ordinary" chain rule- (df/dx)(dx/dy)= df/dy- you are NOT "canceling" the "dx"s.
    Last edited by a moderator: Sep 19, 2009
  4. Sep 19, 2009 #3
    Yes, you are right. i only know how to use chain rule, but i don't know what it is actually.
    When i start to learn Partial Derivative, i find it difficult to understand why dw/dt appear 3 term, and why do dx and dx not be eliminate.

    To understand this problem ,i have read my text and start confusing

    The definition of partial d of z with repect to x at [tex]x_{0},y_{0}[/tex] is
    [tex]f_{x}(x_{0},y_{0})=lim_{x->x0}\frac{f(x,y0)-f(x0-y0)}{x-x0}= \frac {\partialf}{\partialx}[/tex]

    Consider a part of the chain rule, let say : dw/dt= [tex]\frac{\partial w}{\partial x}[/tex] [tex]\frac{dx}{dt}[/tex]

    it is equivalent to [tex]\left[lim_{x->x0}\frac{f(x,y0,z0)-f(x0,y0,z0)}{x-x0}\right][/tex] [tex]\left[lim_{t->t0}\frac{x(t)-x(t0)}{t-t0}\right][/tex]

    [tex]lim_{x->x0} x-x_{0}[/tex] isn't the same as [tex] lim_{t->t_{0}} x(t)-x(t_{0})[/tex] ??
    So i get confusing why i can't eliminate them. or i cannot treat them as a fraction??
    Last edited: Sep 19, 2009
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