Appendix A:
Finding the equation for the derivative

, , , , and are all functions of x.

We know that the normal and the vector tangent to are orthogonal (see Figure 4), meaning that their dot product is equal to zero:

Substituting in for , we get

Since (see Figure 4), and both and are functions of x, . Substituting,

 It is important to realize that , because . If we take the derivative of the equation with respect to x, we get . With these two facts, we may simplify our equation:

We must now replace all occurrences of with and simplify:

Recalling that and , we can further simplify: