Question

if y=(sinx ) ^ x find dy/dx​

Genius challenge

Answer 1

Explanation:

Answer:

( sin x )^x (  x . cot x + ㏑ sin x  )

Step-by-step explanation:

Given :

y = ( sin x )^x

Taking ㏑ both side we get :

= > ㏑ y = ㏑ ( sin x )^x

= > ㏑ y = x . ㏑ sin x

We know :

( ㏑ x )' = 1 / x

( sin x )' = cos x

Diff. w.r.t. x :

= > 1 / y ( d y / d x ) = x ( ㏑ sin x )' + ㏑ sin x ( x )'

= >  1 / y ( d y / d x ) = x ( ㏑ sin x )' + ㏑ sin x

= >  1 / y ( d y / d x ) = x / sin x . ( sin x )' + ㏑ sin x

= >   1 / y ( d y / d x ) = x / sin  x . ( cos x ) + ㏑ sin x

= >  1 / y ( d y / d x ) = x . cos x / sin x + ㏑ sin x

= >  1 / y ( d y / d x ) =  x . cot x + ㏑ sin x

Putting value of y = ( sin x )^x

= > d y / d x = ( sin x )^x (  x . cot x + ㏑ sin x  )

Hence we get required answer!