- #1

drawar

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## Homework Statement

If [itex]f[/itex] is homogeneous of degree [itex]n[/itex], show that [itex]f_{x}(tx,ty)=t^{n-1}f_{x}(x,y)[/itex].

## Homework Equations

## The Attempt at a Solution

There are many solutions out there, and here's one of them:

Since [itex]f[/itex] is homogeneous of degree [itex]n[/itex], [itex]f(tx,ty)=t^{n}f(x,y)[/itex] for all [itex]t[/itex], where [itex]n[/itex] is a positive integer.

Taking the partial derivative wrt [itex]x[/itex]

[itex]\frac{\partial }{{\partial (tx)}}f(tx,ty).\frac{{\partial (tx)}}{{\partial x}} + \frac{\partial }{{\partial (ty)}}f(tx,ty).\frac{{\partial (ty)}}{{\partial x}} = {t^n}\frac{{\partial f(x,y)}}{{\partial x}}[/itex]

[itex] \Rightarrow t{f_x}(tx,ty) = {t^n}{f_x}(x,y)[/itex] and the desired follows.

The proof is nice, but I just don't get it why from step 1 to step 2, [itex]\frac{\partial }{{\partial (tx)}}f(tx,ty) = \frac{\partial }{{\partial x}}f(tx,ty)[/itex] and then it's rewritten as [itex]{f_x}(tx,ty)[/itex]. Any help is very much appreciated, thanks!