# How do you determine (d^2y)/(dx^2) given sinx-4cosy=y?

Differentiate with respect to variable x the given equation

$\left(\sin x - 4 \cos y\right) ' = \left(y\right) '$

$\cos x + 4 \sin y \cdot y ' = y '$

Differentiate again to get

$- \sin x + 4 \cos y \cdot y ' \cdot y ' + 4 \sin y \cdot \left(y ' '\right) = y ' '$

$- \sin x + 4 \cos y \cdot {\left(y '\right)}^{2} = y ' ' \cdot \left(1 - 4 \sin y\right)$

$y ' ' = \frac{- \sin x + 4 \cdot \cos y \cdot {\left(y '\right)}^{2}}{1 - 4 \sin y}$

Note : We suppose that y is a funtion of x hence $y = y \left(x\right)$