(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We say that a differentiable function [tex]f : \mathbb{R}^n \rightarrow \mathbb{R}[/tex] is homogenous of degree p if, for every [tex]\mathbf{x} \in \mathbb{R}^n [/tex] and every a>0,

[tex]f(a\mathbf{x}) = a^pf(\mathbf{x}).[/tex]

Show that, if f is homogenous, then [tex]\mathbf{x} \cdot \nabla f(\mathbf{x}) = p f(\mathbf{x})[/tex] .

2. Relevant equations

The chain rule (not sure if I need it): [tex]\displaystyle \frac{d}{dt} f(\mathbf{x}(t)) = \Sigma_{i = 1}^{n} f_{x_i}\dot{x_i} = \dot{\mathbf{x}} \cdot \nabla f[/tex]

3. The attempt at a solution

Well, I see the resemblance between the rightmost hand side of the chain rule I wrote down, but I don't really understand how the chain rule is applied in this situation, seeing as there isn't anything about x being a function of something else here.

Any ideas?

Cheers.

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# Homework Help: Multivariable Calculus: Applications of Grad (and the Chain Rule?)

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