Why does a homothetic function have constant ratio of marginal products along rays? A homothetic ordering is defined as

$x \succeq y \Rightarrow \lambda x \succeq \lambda y \qquad \forall \lambda &gt;0$

where $x,y \in \mathbb{R}^n$

Then, any differentiable function representing the ordering has the property

$\frac{\partial f}{\partial x_i}(\lambda x)= k \frac{\partial f}{\partial x_i }(x)$

with $k,\lambda &gt;0$

How is this results derived? 

I can see how we can derive properties of the function values from homothety, but have no idea how we can say anything about its derivatives.
A homothetic function can be characterized as follows:
Let $f(\mathbf x)$, $\mathbf x \in \mathbb R^n$ be a function homogeneous of degree $r$. Let $g()$ be a function with $g'\neq 0$. Then 

$$G(\mathbf x) = g[f(\mathbf x)]$$
is homothetic. Since $f(\mathbf x)$ is homogeneous of degree $r$ we have that 

$$f(\lambda \mathbf x) = \lambda ^ rf(\mathbf x) $$

Then 

$$G(\lambda \mathbf x) = g[\lambda ^ r f(\mathbf x)]$$ and so 

$$\frac{\partial G(\lambda x)}{\partial x_i}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_i}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_i}$$

Evidently, we will also have 

$$\frac{\partial G(\lambda x)}{\partial x_j}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_j}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_j}$$

which leads to the "constant MRS along rays" characterization of homothetic functions,

$$\frac{\partial G(\lambda x) / \partial x_i}{\partial G(\lambda x) / \partial x_j} = \frac{\partial G (x) / \partial x_i}{\partial G( x) / \partial x_j}$$

(see Simon and Blume 1994, p. 503).