Answer
Regard $y$ and $z$ and differentiate with respect to $x$:
$$\frac{\partial}{\partial x}w=\frac{y}{\cos^2(x+2z)}.$$
Regard $x$ and $z$ and differentiate with respect to $y$:
$$\frac{\partial}{\partial y}w=\tan(x+2z).$$
Regard $x$ and $y$ and differentiate with respect to $z$:
$$\frac{\partial}{\partial z}w=\frac{2y}{\cos^2(x+2z)}.$$
Work Step by Step
Regard $y$ and $z$ and differentiate with respect to $x$:
$$\frac{\partial}{\partial x}w=\frac{\partial}{\partial x}(y\tan(x+2z))=y\frac{\partial}{\partial x}\tan(x+2z)=y\frac{1}{\cos^2(x+2z)}\frac{\partial}{\partial x}(x+2z)=\frac{y}{\cos^2(x+2z)}.$$
Regard $x$ and $z$ and differentiate with respect to $y$:
$$\frac{\partial}{\partial y}w=\frac{\partial}{\partial y}(y\tan(x+2z))=\tan(x+2z)\frac{\partial}{\partial y}y=\tan(x+2z).$$
Regard $x$ and $y$ and differentiate with respect to $z$:
$$\frac{\partial}{\partial z}w=\frac{\partial}{\partial z}(y\tan(x+2z))=y\frac{\partial}{\partial z}\tan(x+2z)=y\frac{1}{\cos^2(x+2z)}\frac{\partial}{\partial z}(x+2z)=\frac{2y}{\cos^2(x+2z)}.$$
