Verified answer

the cylinder is perpendicular to the z-axis,

so one of these equations is true:

x² + z² = r² (cylinder parallel to y-axis) or

y² + z² = r² (cylinder parallel to x-axis)

for both equations, the third dimension is constant and correlates to the length of the cylinder. r is also constant, as it refers to the radius of the circle that is the cross section of the cylinder.

so, if you take the first partial derivative of either equation, you get z=0