In Calculus textbooks, the concept of curvature is usually presented in the chapter of Polar Coordinates, and the exposition is done, of course, in that context, i.e., that of polar coordinates, using angles. Afterwards usually there is a “translation” to Cartesian coordinates that, I feel, is not very natural. Or, as in the example below, from http://mathworld.wolfram.com/Curvature.html one finds a description using parametric equations.
A simpler, more direct and intuitive approach for begining students may be obtained considering the curvature explicitly as the rate of change of the tangent with respect of a portion of arc. In this sense we could formally say:
As we seek the variation of y´ with respect to s´ we obtain the derivative of C(x), thus:
taking the first derivative, we obtain:
or, after rearranging,
which is that which we were searching.