Suppose we want to find a root of the function
f = @(x) x.*exp(x) - 2;
fplot(f,[0 1.5])
xlabel('x'), ylabel('y') % ignore this line
set(gca,'ygrid','on') % ignore this line
title('Objective function') % ignore this line
From the graph, it is clear that there is a root near 
. So we call that our initial guess, 
.
. So we call that our initial guess, .x1 = 1;
f1 = f(x1)
hold on, plot(x1,f1,'.')
Next, we can compute the tangent line at the point 
, using the derivative.
, using the derivative.dfdx = @(x) exp(x).*(x+1);
slope1 = dfdx(x1);
tangent1 = @(x) f1 + slope1*(x-x1);
fplot(tangent1,[0 1.5],'--')
title('Function and tangent line') % ignore this line
In lieu of finding the root of f itself, we settle for finding the root of the tangent line approximation, which is trivial. Call this 
, our next approximation to the root.
, our next approximation to the root.x2 = x1 - f1/slope1
plot(x2,0,'.')
f2 = f(x2)
title('Root of the tangent') % ignore this line
The residual (value of f) is smaller than before, but not zero. So we repeat the process with a new tangent line based on the latest point on the curve.
cla, fplot(f,[0.8 0.9])
plot(x2,f2,'.')
slope2 = dfdx(x2);
tangent2 = @(x) f2 + slope2*(x-x2);
fplot(tangent2,[0.8 0.9],'--')
x3 = x2 - f2/slope2;
plot(x3,0,'.')
f3 = f(x3)
title('Next iteration') % ignore this line
We appear to be getting closer to the true root each time.