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Differentiate (sinx)^(cosx) with respect...

Differentiate `(sinx)^(cosx)` with respect to `x` :

Text Solution

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Let `y = (sinx)^(cosx)`
`rArr logy = log(sinx)^(cosx)= cosxlogsinx`
`:. (d)/(dy) logy.(dy)/(dx)=(d)/(dx)(cosx.logsinx)`
`rArr (1)/(y). (dy)/(dx) = cosx . (d)/(dx) log sinx + logsinx.(d)/(dx) cosx`
`= cosx. (1)/(sinx).(d)/(dx)sinx+logsinx.(-sinx)`
`= cotx. cosx - log(sinx).sinx, [:' cotx = (cosx)/(sinx)]`
`:. (dy)/(dx) = y[(cos^(2)x)/(sinx)-sinx.log(sinx)]`
`= sinx^(cosx)[(cos^(2)x)/(sinx)-sinx.log(sinx)]`
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