I was wondering how radius of curvature was derived, and this is what I came up with. It turned out to be longer than expected. Then I looked at how it compares with other (presumably more mathematically accurate) derivations on the Internet. Everywhere I looked had the same approach, where they first looked at curvature by finding the rate of change of the angle between the tangent at the point of interest and the $x$-axis, with respect to arc length, and then getting to the radius of curvature. What I've done seems more of a direct approach to me, maybe more geometric as well.
What I started with:
What I finished with:
$$R = \left | \frac{(1+f'(a)^2))^{\frac{3}{2}}}{f''(a)} \right |$$
Before getting to that I made these statements:
$$\left ( \lim_{h\rightarrow 0}f(a+h)=f(a)\right )$$
and \begin{align*} \text{By definition:}\qquad \lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h} &= f'(a), \\ \text{and}\qquad \lim_{h\rightarrow 0} \frac{f'(a+h)-f'(a)}{h} &= f''(a). \end{align*}
I'm not sure about the statement in the brackets on the fourth page. It feels right, but I'm not sure. That statement also seems like it's contradicting the "By definition..." statement on the third page? Any thoughts on this, or on the derivation in general?
(Full derivation at https://drive.google.com/file/d/0B4LamLT1ywM1M1NzOHVUQnFvd2M/view?usp=sharing&resourcekey=0-0wvgtszqdJPteKUxY3Fakg)
Update: Changed link, also by "It turned out to be longer than expected" I mean 4 pages long...