Your last line use fixed values, so it will only work for one case. Additionally there was an error in the order of the variable assignment on the loop. Following code should work:
x=0;
xold=inf;
i=0;
format short
while abs(x-xold)>.01
xold = x; % This should be done before the calculation, otherwise you have only 1 iteration
f = sin(x)-cos(x);
f_1 =cos(x)+sin(x);
x = x - f/f_1; % Note f and f_1 instead of y and dy
i = i+1;
end
figure
xplot = linspace(-8,8,100);
plot(xplot,cos(xplot)),hold on
plot(xplot,sin(xplot))
plot([x], cos(x), '*r' )
plot([x], sin(x), '*r' )
