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Find the derivative of y = cos(x)^(sin(x))

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In normal derivation we know that;

`(d(x^n))/dx = nx^(n-1)`

This is valid only when n is a constant or n is not a function of x.

 

For our question we cannot use the above rule because sinx is not a constant.
So we have to use logarithm method to solve this.

`y = cos(x)^(sin(x))`

Take log in both sides.

`logy = log(cos(x)^(sin(x)))`

`logy = sinxlogcosx`

Get the derivative on both sides.

`1/y*(dy)/dx = sinx*1/cosx*(-sinx)+logcosx*cosx`

     ` (dy)/dx = y(cosxlogcosx-(sin^2x)/cosx)`

 

  `(dy)/dx = (cos(x)^(sin(x)))(cosxlogcosx-(sin^2x)/cosx)`

 

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