implicit differentiating equation with $\cos$

Question

I need help getting $\frac{d^2y}{dx^2}$ for $y−\cos y=2x$

Someone answered and got $(1+\sin y(x))3+4\cos y(x)$ but i was unable to follow their steps and didnt get how to do it. any HELP?

Answer 1

$$y(x)−\cos(y(x))=2x$$ $$y'(x)+\sin(y(x))y'(x)=2$$ $$\cdots$$

Answer 2

Hint

Let us consider the implicit function $$f=y−\cos y-2x=0$$ So, we have $f'_x=-2$ and $f'_y=1+\sin(y)$. So, as usual from implicit differentiation, $$y'_x=\frac {2}{1+\sin(y)}$$ Differentiating both sides with respect to $x$, we then have $$y''_x=- \frac {2 y'_x \cos(y)}{(1+\sin(y))^2}$$ Replace $y'_x$ by its expression.

I am sure that you can easily take from here.