What you appear to be describing is the derivation of a linear least square regression. In that context, ‘linear’ implies ‘linear in the parameters’, such the partial derivatives of the objective function with respect to each parameter are not functions of that parameter or of any other parameters.
Your data are best determined by fminsearch. There are other options, however those require the Optimization Toolbox, the Statistics and Machine Learning Toolbox or Curve Fitting Toolbox.
Example —
xv = 1:10;
yv = sqrt(xv);
objfcn = @(b,x) b(1).*(1-exp(-b(2).*x));
B0 = rand(2,1);
[B,fval] = fminsearch(@(b)norm(yv-objfcn(b,xv)), B0)
figure
plot(xv,yv,'x', 'DisplayName','Data')
hold on
plot(xv, objfcn(B,xv), '-r', 'DisplayName','Regression Fit')
hold off
grid
xlabel('x')
ylabel('y')
legend('Location','best')
text(1,3, sprintf('$y = %.2f\\ (1-e^{-%.2f \\ x})$',B), 'Interpreter','latex')
EDIT — (17 Feb 2023 at 16:58)
Corrected typographical errors.
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