设 $f(x, y, z)$ 是 $k$ 次齐次函数, 即 $f(t x, t y, t z)=t^k f(x, y, z), \lambda$ 为某一常数, 则结论正确的是 ( )
A. $x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}=k^\lambda f(x, y, z)$
B. $x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}=$ $\lambda^k f(x, y, z)$
C. $x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}=k f(x, y, z)$
D. $x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}=$ $f(x, y, z)$