Answer
$$\frac{\partial}{\partial x}w=\frac{1}{x+2y+3z};$$
$$\frac{\partial}{\partial y}w=\frac{2}{x+2y+3z};$$
$$\frac{\partial}{\partial z}w=\frac{3}{x+2y+3z}.$$
Work Step by Step
Regard $y$ and $z$ as constant and differentiate with respect to $x$:
$$\frac{\partial}{\partial x}w=\frac{\partial}{\partial x}\ln(x+2y+3z)=\frac{1}{x+2y+3z}\frac{\partial}{\partial x}(x+2y+3z)=\frac{1}{x+2y+3z}\cdot(1+0+0)=\frac{1}{x+2y+3z}.$$
Regard $x$ and $z$ as constant and differentiate with respect to $y$:
$$\frac{\partial}{\partial y}w=\frac{\partial}{\partial y}\ln(x+2y+3z)=\frac{1}{x+2y+3z}\frac{\partial}{\partial y}(x+2y+3z)=\frac{1}{x+2y+3z}\cdot(0+2+0)=\frac{2}{x+2y+3z}.$$
Regard $x$ and $y$ as constant and differentiate with respect to $z$:
$$\frac{\partial}{\partial z}w=\frac{\partial}{\partial z}\ln(x+2y+3z)=\frac{1}{x+2y+3z}\frac{\partial}{\partial z}(x+2y+3z)=\frac{1}{x+2y+3z}\cdot(0+0+3)=\frac{3}{x+2y+3z}.$$
