If `y=f(a^x)a n df^(prime)(sinx)=(log)_e x ,then f i n d ((dy)/(dx)),` if it exists, where pi/2 less than x less than pi

Explanation

Given that `f'(sin x)=(df(sin x))/(d(sin x)=log_(e) x`
`=log_(e)(pi-sin^(-1)(sin x))`
`[because sin^(-1) sin x = pi -x x for x in (pi//2,pi)]`
`(dt(t))/(dt)=log_(e)(pi-sin^(-1)t)`
`"or "f'(t)=log_(e)(pi-sin^(-1)t)" (1)"`
`and " "y=f(a^(x))`
`therefore" "(dy)/(dx)=f'(a^(x))a^(x)log_(e)a`
`=a^(x)log_(e)a log_(e)(pi-sin^(-1)a^(x))" [using (1)]"`