Get derivative numerically

Question

I have 2 vectors and a scalar:

• grid which is (N x 1);
• Value which is (N x 1);
• sval which is (1,1);

If I want to interpolate sval on grid I know that I can simply do:

intervalue = interp1(grid, Value, sval, 'PCHIP');

What if now I want the derivatives, i.e. the slope of the function Value at that particular point sval?

Answer 1

As mentioned in the comments, you can approximate the derivative by forward finite difference approximation:

slope = diff(Value) ./ diff(grid);

Alternatively:

slope = gradient(Value(1:end-1),grid);

This is a simple method of numerical differentiation. For a detailed guide on numerical differentiation in MATALB, see this answer.

Here is an example of the finite difference method with the desired interpolation:

% Define function y = x^3
grid = 1:100;
Value = grid .^ 3;

% Approximate derivative via the local slope
slope = diff(Value) ./ diff(grid);
% Or: slope = gradient(Value(1:end-1),grid);
slope_grid = grid(1:end-1);

% Interpolate derivative
sval = 33.5;
sval_slope = interp1(slope_grid, slope, sval, 'PCHIP');

We can visualize the result:

figure;
plot(grid, 3*grid.^2)
hold on
plot(slope_grid, slope)
legend('Reference', 'Approximation', 'Location', 'NorthWest')