If V is a homogeneous function of degree $n$ , how is $\frac{\delta V}{\delta x}$ a homogeneous function of degree $(n-1)$

Question

I need to prove the above. I can show it by taking a polynomial $f(x, y, z)$ of degree $n$ as $V$ and $\frac{\delta V}{\delta x}$ will obviously have degree $(n-1)$. But I want to prove it properly, not using examples. I took $V = x^n \phi(\frac{y}{x}, \frac{z}{x})$ .

$\frac{\delta V}{\delta x}$ = $x^n \frac{\delta \phi}{\delta x} + nx^{n-1}\phi$

I don't know how to proceed from here. Any ideas?

Answer 1

Use definition of derivative, $$\lim_{\delta x \rightarrow 0} \frac{V(x+\delta x) - V(x)}{\delta x}.$$ Expand using Newton's binomium. You should find that the term in $x^n$ cancels but $x^{n-1}$ remains in the limit.

Answer 2

Suppose $$f(x,y,z)=\sum_{a+b+c=n}{C_{abc}x^ay^bz^c}$$ Then $$\frac{\partial f}{\partial x}=\sum_{a+b+c=n,\ a\neq 0}{C_{abc}x^{a-1}y^bz^c}$$ Notice that $(a-1)+b+c=n-1$, so $\frac{\partial f}{\partial x}$ is homogeneous of degree $n-1$.