As part of a proof of Euler's homogeneous theorem and a function $g(\lambda x,\lambda y,z)$, the author differentiates wrt $\lambda$ and obtains
$$ \frac{\partial}{\partial \lambda}g(\lambda x, \lambda y, z ) = g_x(\lambda x, \lambda y, z) \cdot x + g_y (\lambda x, \lambda y, z) \cdot y $$
Where $g_x$ and $g_x$ are the partial derivatives of $g$ wrt $x$ and $y$.
Can someone explain why one is allowed to do this?
This is just the chain rule: $$\frac{\partial}{\partial \lambda} g(\lambda\cdot x,\lambda \cdot y,z)=\frac{\partial g}{\partial x}{\frac{\partial x}{\partial \lambda}}+\frac{\partial g}{\partial y}\frac{\partial y}{\partial \lambda}+\frac{\partial g}{\partial z}\frac{\partial z}{\partial \lambda}$$.
Note $\frac{\partial z}{\partial \lambda}=0$.