So I am using Newton's method. And for equation $f(x)=\sin(x)/(3+\sin(x))$ I differentiated it and I got $3\cos(x)/(3+\sin(x))^2$ which is checked by wolfram alpha. However, when I apply Newton's method on my own my results do not match my expected result calculated by a Newton's method software which is shown below 
The picture is what I am aiming, but it seems like there are some errors.
. What is the error? I really could not find it.
Justin Chan St georges school IB mathematics EE
Since you do not show us your work it is difficult to find the error. It must be either in the function that you must iterate, which should be $$ N(x)=x-\frac{f(x)}{f'(x)}=x-\frac13\,(3+\sin x)\tan x, $$ or in the computations with that function. The results given by the software are correct.