I am trying to understand how the curvature equation
$\kappa = -\frac{f_{xx} f_y^2-2f_{xy} f_x f_y + f_x^2 f_{yy}}{(f_x^2+f_y^2)^{3/2}}$
for implicit curves is derived. These curves arise from equalities such as $f(x,y)=0$. I found this on the net:
http://www.cad.zju.edu.cn/home/zhx/GM/001/00-rep_dg.pdf
I can follow almost everything here until pg 49, then the author jumps to the final equation and I have no idea how he's done it.
Can anyone help, or point to other possible derivations? I understand the parametric form of curvature equation which is $\kappa = | \frac{d\vec{T}}{ds} |$ where $\vec{T}$ is unit tangent, if any parallels need to be made to that subject, just in case.
And one more question: How do I expand the term below?
$\frac{\partial}{\partial x} \bigg( \frac{f_y}{\sqrt{f_x^2 + f_y^2}} \bigg)$
Do I have to use the Quotient Rule?
$\frac{d}{dx}(\frac{u}{v}) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$
and in that case, I guess I would need to derive $\frac{\partial}{\partial x}(\sqrt{f_x^2+f_y^2})$. Would this be $\frac{1}{2}\frac{2f_x f_{xx} + 2f_y f_{yx}}{\sqrt{f_x^2+f_y^2}}$
Thanks again