Rewriting the derivative of a homogenous function demonstration

[bobbarker]

Homework Statement

Suppose that f=f(x₁,x₂,...,x_n) is a homogeneous function of degree r with mixed partial derivative of all orders. Show that

Can this be generalized?

Homework Equations

We say that a function f is homogeneous of degree r if there exists r such that f(tx₁,tx₂,...,tx_n) = t^r * f(x₁,x₂,...,x_n)

The Attempt at a Solution

I'm pretty lost on how to show these two. I understand that this is kind of like taking the "second derivative" of f with respect to t, but how do I introduce the t into the sum terms?

 

[grey_earl]

We say that a function f is homogeneous of degree r if there exists r such that f(tx₁,tx₂,...,tx_n) = t^r * f(x₁,x₂,...,x_n)

Just start from this equation and derive both sides with respect to t, then set t = 1. The generalization to all derivatives is obvious to you, isn't it?