I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation:
$$\cos\cos(x^3y^2) - x \cot y = -2y$$
I figured that the calculation requires the chain rule to differentiate the composite function, but I'm not sure how to 'remove' the y with respect to x from inside the composite function. My calculations are:
$$\frac{dy}{dx}[\cos\cos(x^3y^2) - x \cot y] = \frac{dy}{dx}[-2y]$$
$$\frac{dy}{dx}[\cos\cos(x^3y^2)] = \sin \cos (x^3y^2 \cdot y'(x)) \cdot \sin (x^3y^2 \cdot y'(x)) \cdot 6x^2y\cdot y'(x)$$
This seems a bit long and convoluted. I'm also not sure how this will allow me to solve for $y'(x)$. Carrying on...
$$\frac{dy}{dx}[x \cot y] = -\csc^2y \cdot y'(x)$$
$$\frac{dy}{dx}[-2y] = -2$$
Is my calculation correct so far? This seems to be a very complex derivative. Any comments or feedback would be appreciated.