Why Does the Multidimensional Chain Rule Lead to Euler's Theorem Proof?

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Homework Statement
Proof ##\sum\limits_{i=1}^{n} x_i \frac{\partial f}{\partial x_i}(x)=n f(x)##
Relevant Equations
Chain rule
Hi,

I am having problems with the following task:

Bildschirmfoto 2024-06-13 um 14.38.20.png

My lecturer gave me the tip that I should apply the multidimensional chain rule to obtain the following transformation ##\sum\limits_{i=1}^{n} \frac{\partial f}{\partial x_i}(x) \cdot x_i= \frac{d f(tx)}{dt} \big|_{t=1}##

Unfortunately, I don't know why the two terms should be equal because of the chain rule. There is a sum on the left and a derivative on the right side.

One more thing, in the 1D case the chain rule is ##[f(g(x))]'=f'(g(x)) \cdot g'(x)## but the term ##g'(x)## for the multidimensional case is missing on the right-hand side.
 
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You have two ways of calculating \frac{d}{dt}f(tx). One is to apply the chain rule, as you would for any function: <br /> \frac{d}{dt}f(tx) = \sum_i \frac{\partial f}{\partial x_i} \frac{d}{dt}(tx_i). The other is to use the particular feature of f that f(tx) = t^nf(x): <br /> \frac{d}{dt}f(tx) = \left(\frac{d}{dt} t^n\right)f(x). These two expressions must be equal to each other. What then happens if you evaluate the derivatives and set t = 1?
 
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Thank you pasmith for your help 👍


I think I'm beginning to understand the derivation now. I was also wondering all the time how the ##t## gets into the equation, but since you evaluate the derivative for ##t=1##, you can claim that the two derivatives are equal.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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