# Infinitesimal length in the cylindrical system

I understand how to use the distance formula to arrive at this:

"An infinitesimal length in the rectangular system is given by $dL^2= dx^2+dy^2+dz^2$ ."

How do you prove the following?

"In the cylindrical system the corresponding quantities are $dL^2= dr^2+r^2dφ^2+dz^2$ ."

I think you mean to put $$dL^2=dx^2+dy^2+dz^2\\dL^2=dr^2+r^2dφ^2+dz^2$$ For your proof, in a cylindrical coordinate system $$r^2=x^2+y^2\\x=rcos(φ)\\y=rsin(φ)\\z=z$$ and because of product rule $$\\dx=d(rcos(φ))=dr*cos(φ)+r*d(cos(φ))$$ and $$\frac{d}{dφ}cos(φ)=-sin(φ)\\d(cos(φ))=-sin(φ)dφ$$ it follows that $$\\dx=cos(φ)dr-rsin(φ)dφ$$
The expression $$d(f(x)g(x))=f(x)*dg(x)+g(x)*df(x)$$