Homogeneous function differential degree

Question

Let $f:\mathbb R^n \to \mathbb R$ be a homogeneous function of degree $p$. $f(tx)=t^pf(x)$ for $p>0$.

Let us assume that $f$ is continuously differentiable. How to show that $\frac{\partial f}{\partial x_i}$ is a homogeneous function of degree $p-1$?

I tried to use euler's theorem $(\nabla f(x),x)=kf(x)$ but didn't manage to prove it.

Thanks!

Answer 1

By the chain rule:

$$\frac{\partial}{\partial x_i}\Big[f(tx)\Big]=f_i(tx)\cdot t$$

and

$$\frac{\partial}{\partial x_i}\Big[t^pf(x)\Big]=t^pf_i(x)$$

Can you conclude?