Expert: Paul Klarreich - 8/21/2009

Question
Show that if z=x^nf(u), where u=y/x, then x∂z/∂x + y∂z/∂y = nz

Answer
Questioner: CW
Country: Singapore
Category: Calculus
Private: No
Subject: Calculus
Question: Show that if z=x^nf(u), where u=y/x, then x∂z/∂x + y∂z/∂y = nz

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Hi, CW.

First, a WARNING: The special partial-derivative symbols might not come through the editing process.  You will have to cope.

∂z/∂x = nx^(n-1) f(y/x) + x^n f'(y/x)[- y/x^2]

∂z/∂x = nx^(n-1) f(y/x) - x^(n-2) f'(y/x)[y]

x ∂z/∂x = nx^n f(y/x) - x^(n-1) f'(y/x)[y]

Now
∂z/∂y = x^n f'(y/x) [1/x]

∂z/∂y = x^n-1 f'(y/x)

y ∂z/∂y = x^n-1 y f'(y/x)

The sum is : nx^n f(y/x) - x^(n-1) f'(y/x)[y] + x^n-1 y f'(y/x)

and the last two terms cancel.

True enough.