Coordinate systems/Derivation of formulas/Cylindrical from Cartesian unit vectors

From Wolfram Mathworld, we see the following relations:

${\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\cos \phi {\hat {\mathbf {x} }}+\sin \phi {\hat {\mathbf {y} }}\\{\hat {\boldsymbol {\phi }}}&=-\sin \phi {\hat {\mathbf {x} }}+\cos \phi {\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$

Additionally, it has already been shown on the same page that:

${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\x&=\rho \cos \phi \\y&=\rho \sin \phi \\\end{aligned}}$

Now, we simply need to combine these relations together to obtain:

${\begin{aligned}\cos \phi &={\frac {x}{\sqrt {x^{2}+y^{2}}}}\\\sin \phi &={\frac {y}{\sqrt {x^{2}+y^{2}}}}\\\end{aligned}}$

And hence:

${\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x}{\sqrt {x^{2}+y^{2}}}}{\hat {\mathbf {x} }}+{\frac {y}{\sqrt {x^{2}+y^{2}}}}{\hat {\mathbf {y} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\phi }}}&=-{\frac {y}{\sqrt {x^{2}+y^{2}}}}{\hat {\mathbf {x} }}+{\frac {x}{\sqrt {x^{2}+y^{2}}}}{\hat {\mathbf {y} }}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$

As was to be demonstrated.