Euler's Theorem Proof Inquiry

Question

I am having trouble understanding the following equation (appears in the proof for Euler's theorem for homogeneous functions).

More specifically: I do not understand this chain rule. How is it that when you take the derivative w.r.t. $k$, on the RHS we get partials w.r.t. $kx_1$, $kx_2$, etc..

Can someone perhaps give an example of how that is possible?

I should add for clarity: $\bar x = x_1, x_2,...,x_N$

Answer 1

Hint:

$$f(kx)=f([kx_1,kx_2,...,kx_N]^T)=f(kx_1,...,kx_n)$$

The chain rule is given by

$$\dfrac{df}{dk}=\sum_{n=1}^{N}\dfrac{\partial f}{\partial kx_n}\dfrac{\partial kx_n}{\partial k}.$$

The chain rule means that you first differentiate with respect to the argument and then multiply by the differentiation of the argument with respect to $k$. As you have $N$ arguments you need to add $N$ such terms.