I am trying to derive $e^x \sin x - 2x \csc x$. I tried using the product and difference rule. So I got the derivative for $e^x \sin x$ and got $(e^x)(\cos(x))+(\sin(x))(e^x)$ and for the derivative of $2x \csc x $ I got $(2(x))(-\csc(x) \cot(x))+2\csc(x)$. So now I used the difference rule and simplified and what not and I ended with $(e^x)(\cos(x)+\sin(x))-2x(-\csc(x)\cot(x))+2\csc(x)$. When I entered the answer it was wrong. The answer is $(e^x)(\cos(x)+\sin(x))+2(x \cot(x)-1)\csc(x).$
I would really appreciate if someone could point out what I did wrong and how to get the correct answer, step by step preferably so I know how to deal with this issue. All help is appreciated!