# Euler´s theorem [duplicate]

This question already has an answer here:

With the follow definition

Definition: We say that the function $$f(x_1, \cdots ,x_n)$$ is positive-homogeneous of degree $$k$$ in $$x_1, \cdots, x_n$$ if
$$f(\lambda x_1, \cdots, \lambda x_n) = \lambda^kf(x_1, \cdots ,x_n)$$ for every $$\lambda > 0$$

Prove the Euler's theorem

If $$f(x_1, \cdots ,x_n)$$ is continuously differentiable and positive-homogeneous of degree k, then
$$\sum_{i=1}^n \frac{\partial f}{\partial x_i} x_i = kf$$

I don't need a complete proof, I only need a any hint.

I appreciate any help

## marked as duplicate by Henning Makholm, user10354138, Dietrich Burde, Lee David Chung Lin, YuiTo ChengJun 27 at 0:17

Hint: Differentiate with respect to $$\lambda$$ in $$f(\lambda x_1,\lambda x_2,\dots,\lambda x_n)=\lambda^k f(x_1,x_2,\dots,x_n).$$
• ...and then choose and substitute a special value of $\;\lambda\;$.... – DonAntonio Jun 26 at 17:51