[Math] Help to understand the proof of partial derivatives of homogeneous functions

partial derivativeproof-writing

I found this short proof that says the partial derivaties of homogenous functions of degree $k$ is homogeneous of degree $k-1$. Here is the proof in its entirety:

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I am lost at the very first step of the proof which says to differentiate with respect to $x_i$ both sides of the equation:

$f(tx_1,tx_2,\dots,tx_n)=t^kf(x_1,x_2,\dots,x_n)$

Based on my understanding of partial derivates, if I were to differentiate the left hand side, I will get this:

$tf'_i(x_1,x_2,\dots,x_n)$

Which is not the same as what the proof says it should be.

Please advise.

Best Answer

To be sure, do each step carefully. Write $$ \phi_t(x)=(tx_1\ldots tx_n).$$ Therefore you are trying to compute $$\frac{\partial}{\partial x_i} \left( f\circ \phi_t\right)(x_1\ldots x_n).$$ Now use the chain rule.

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