It's because of calculation first and last point. gradient does it this way:
first_grad = (y2-y1)/dx;
second_grad = 1/2*(y4-y3)/dx + 1/2*(y3-y2)/dx;
third_grad = 1/2*(y5-y4)/dx + 1/2*(y4-y3)/dx;
%% ...
last_grad = (ylast-ylast_1)/dx;
BLack curve and black points is your original data. Red points are points where gradient/derivative is calculated. Pay attention that first and last points are not at the beginning and end
Second time you use gradient you use the same step (again). But for the first and last points step/2 should be used.
Interesting fact
If you have not regular step (h is not equal) gradient can't handle it. Better use diff
% generate some data
x = sort(10*rand(100,1));
y = sin(x);
dy1 = gradient(y,x);
dy2 = diff(y)./diff(x);
x2 = x(2:end)-diff(x)/2; % actual x for diff/derivative
plot(x,y) % original data
hold on
plot(x,dy1,'g') % derivative obtained with gradient
plot(x2,dy2,'.m') % derivative obtained with diff
plot(x,cos(x),'k') % exact derivative
hold off
legend('orignal data','gradient','diff','exact')
