Q:

Derivative of $y-2\sin(y)=x$ defines: $y=y(x)$

I need to find the derivative of $y'$ and $y''$ given that:
$$y-2\sin(y)=x$$
defines:
$$y=y(x)$$
Thank you!

A:

HINT:
Differentiating both sides of the given equation with respect to $x$ reveals
$$y'(x)-2\cos(y)y'(x)=1 \tag 1$$
SPOLIER ALERT:  Scroll over the highlighted area to reveal the solution.

Solving $(1)$ for $y'(x)$ yields $$y'(x)=\frac{1}{1-2\cos(y)} \tag 2$$Now, differentiating $(2)$ with respect to $x$ yields $$\begin{align}y''(x)&=-\frac{2\sin(y)y'(x)}{(1-2\cos(y))^2}\\\\&=-\frac{2\sin(y)}{(1-2\cos(y))^3}\\\\&=\frac{x-y}{(1-2\cos(y))^3}\end{align}$$ 

