This is a question that gets aslked many many times about splines. And, well, you can't easily get a simple equation, as there is no one equation. Depending on how many breaks there are in the spline, you may have dozens, or thousands of different "equations", or more. A spline is composed of pieces of polynomial segments, one between each pair of breaks. nd even then it gets a little tricky. So I'm sorry to say it is rarely of any value to try to write down "the" equation.
Instead, you can use tools like fnder to differentiate the spline for you, creating a new spline that can then be evaluated to produce the derivative at any point.
For example...
spl = spline(x,y)
spl =
form: 'pp'
breaks: [0 0.8976 1.7952 2.6928 3.5904 4.4880 5.3856 6.2832]
coefs: [7x4 double]
pieces: 7
order: 4
dim: 1
fplot(@(X) fnval(spl,X),[0,2*pi])
title 'Data, with an interpolating spine through it'
Now differentate the spline, creating a new spline curve that is the derivative of spl. It will be a lower order curve.
splder = fnder(spl)
splder =
form: 'pp'
breaks: [0 0.8976 1.7952 2.6928 3.5904 4.4880 5.3856 6.2832]
coefs: [7x3 double]
pieces: 7
order: 3
dim: 1
As you can see, MATLAB knows the derivative of a spline that was originally cubic (degree == 4) is now made from quadratic segments.
We can now evaluate this derivative at any point.
fplot(@(X) fnval(splder,X),[0,2*pi])
title 'Derivative of the spline'
And you should see this will look a lot like a sine wave. Actually, -sin(x), to be pedantic. Not perfect, since we had only 8 data points.
Finally, yes, you can extract those segments. My SLM toolboix, on the file exchange, as a function that will provide to you those segments as polynomials, if you desperately wanted, The function is called SLMPAR.
C = slmpar(spl,'symabs');
0.14827585699644890704362865108124*x^3 - 0.69064468900003539442167266315664*x^2 + 0.080993821270724186689449197729118*x + 1.0
And so we see the cubic polynomial produced in the fiirst segment of the spline. You will need to download my SLM toolbox of course to use it. Easier just to use fnder.
