If `y=(sin x) ^(sin x ^sin x ....oo)` then prove that `dy/dx=(y^2 cot x)/(1-y log (sin x))`

Explanation

`y=(sin x) ^(sin x ^sin x ....oo)`
`rArr y=(sin x)^y `
`rArr log y =log (sin x)^y=y log (sin x)`
Differentiate both sides with respect to x
`(1)/(y)dy/dx =y.1/sin x .d/dx sin x +log (sin x) .dy/dx `
`rArr dy/dx (1/y-log sin x )=y/sin x .cos x`
`rArr dy/dx (1-y log sin x)/(y)=y cot x `
`rArr dy/dx =(y^2 cot x)/1- y log sin x`