There are two approaches that I know of. You may like to fill in the details of the proofs. One way is to use the relation

R=1/k = ds/dy

where s is the arc length, y is the angle between the positive x-axis and the tangent. Using the chain rule and the fact that tany = dy/dx, ds² = dx² +dy², we get the required result.

The other way is a work directly from the definition. The circle of curvature is the circle which has y, dy/dx and d² y/dx² all agreeing with our curve at the given point. Now from the general equation of circle and these conditions, we can derive the radius of the circle, i.e. the radius of curvature.

k is called the curvature, defined as the rate at which direction of the tangent changes along the curve. To see why R=ds/dy, consider a circle of radius R, we have s=Ry providing one measures s from (0,-R).

The second way is very messy. WLOG, let us assume our point of interest is the origin, and our curve has y ' =p, y ' ' =q at the origin. We have the general equation of circle

(X-a)² +(Y-b)² = R²

Differentiating, we get

dY/dX=(a-X)/(Y-b)=-a/b at (0,0)

d² Y/dX² = -[(X-a)² +(Y-b)² ]/(Y-b)³ = (a² +b² )/b³ at (0,0).

Now force the conditions:

(1) a² +b² = R²

(2) -a/b=p

(3) (a² +b² )/b³ = q

and check all these conditions are satisfied if R=(1+p² )^3/2/q