If `y=(sinx)^((sinx)^((sinx)^(...^infty)))` , prove that `(dy)/(dx)=(y^2cosx)/((1-ylogsinx)`

Explanation

`y=(sin x)^{(sin x)^{(sin x)^{circ}}}`


`therefore y=sin x^{y}`


`log y=log ((sin x)^{y})`.


`log y=y log sin x`.


`frac{1}{y} cdot frac{d y}{d x}=frac{d y}{d x} log sin x+frac{y}{sin x} cos x`


`frac{d y}{d x}(frac{1}{y}-log sin x)=y cot x`.


`frac{d y}{d x}(frac{1-y log sin x}{y})=y cot x`


`therefore frac{d y}{d x}=frac{y^{2} cot x}{1-y log sin x} .`