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Posted by on Sunday, October 5, 2008 at 3:52pm.

Let Z = f(u,v) where u=xy^3 + e^3 and v = (x^2)y.

Obtain expressions for Zx and Zy in terms of fu and fv.

I don't understand the "in terms of" part. How do I do this question? thank you

  • Calculus-partial derivatives - , Sunday, October 5, 2008 at 8:35pm

    dZ = Zx dx + Zy dy (1)

    Also:

    dZ = df = fu du + fv dv (2)

    Express du and dv in terms of dx and dy:

    du = ux dx + uy dy (3)

    dv = vx dx + vy dy (4)


    Substitute (3) and (4) in (2):


    dZ = fu (ux dx + uy dy ) + fv (vx dx + vy dy) =

    (fu ux + fv vx) dx + (fu uy + fv vy)dy


    Comparing this to (1) then gives:

    Zx = fu ux + fv vx

    Zy = fu uy + fv vy

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