[Math] Inverse function of $y=2x+\sin x$

Question

I was doing a long exercise when come to this point: calculate the inverse function of $y=2x+\sin x (x \in\mathbb R) $ and its derivative.
I know that the derivative of an inverse function is $1/f'(x)$ but it is not enough as $x=f^{-1}(y)$. So I tried to find the inverse function but I'm completely stuck just in this point.
Can somebody help me please?
Thanks in advance for your help!!

Answer 1

Hint: write $x=2y+\sin(y)$, then $1=2y'+\cos(y)\cdot y'$, hence $y'=1/(2+\cos(y))$, that will be the derivative of the inverse. If you can solve that differential equation you'll have the inverse ...