Using Newton's Method with a Piecewise Defined Function Using a Natural Cubic Spline approximation, I've generated an approximation polynomial to six points of data. Using the Cubic Spline approximation polynomial, I now need to use Newton's method to find a root of it (the spline approximation). I'm unsure of how to tweak the Newtonian algorithm to handle a piecewise defined function. Any ideas on how to do this?

Hint
The simplest way, at least in my mind, would be to numerically evaluate the derivative by one side difference or by central difference. Select a sufficiently small step size (say $\Delta x=\frac{x}{1000}$). Since your function is continuous, even if piecewise defined, this should not make any problem.
