How to numerically differentiate provided data?

I have the measurements of x with corresponding y displacement lengths:

x = [0,0.375,0.75,1.125,1.5,1.875,2.25,2.625,3];
y = [0,-0.2571,-0.9484,-1.9689,-3.2262,-4.6414,-6.1503,-7.7051,-9.275];

and dy/dx = theta(x) >> theta is the slope

My question is how to numerically differentiate the provided data for displacement y(x)

and the hint is I can choose formulas of any error order.

It's a question combined both math and numerical method, can any one give some help?

Answer 1

[Walter Roberson]

x = [0,0.375,0.75,1.125,1.5,1.875,2.25,2.625,3];
y = [0,-0.2571,-0.9484,-1.9689,-3.2262,-4.6414,-6.1503,-7.7051,-9.275];
theta = gradient(x,y)

theta = 1×9

-1.4586 -0.7908 -0.4381 -0.3293 -0.2806 -0.2565 -0.2448 -0.2400 -0.2389

plot(x, y, x, theta)
legend({'x', 'theta'})

[Lingbai Ren]

Hi Walter, thanks for answering my question! I never used a gradiant function before, but the question mentioned choosing formulas of any error order, do you have any idea what that is or how to assess the problem that way? (We were learning Taylor's series and all kinds of errors in the past week)

[Walter Roberson]

x = [0,0.375,0.75,1.125,1.5,1.875,2.25,2.625,3];
y = [0,-0.2571,-0.9484,-1.9689,-3.2262,-4.6414,-6.1503,-7.7051,-9.275];
orders = 1:7;
for order = orders
   p = polyfit(x, y, order);
   predict = polyval(p, x);
   plot(x, predict, 'displayname', "order = " + order);
   hold on
   err(order-min(orders)+1) = sum((predict - y).^2);
end
xlabel('x'); ylabel('y predicted')
legend show

figure(2)
plot(orders, err);
xlabel('polynomial order'); ylabel('sse')

[Lingbai Ren]

Thanks Walter! Sorry for the late reply!