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

Differentiate `(logx)^(cosx)` with respect to `xdot`

Text Solution

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`"Let "y=(log x)^(cos x).`
Taking logarithm on both the sides, we obtain
`log y = cos xcdot log (log x)`
Differentiating both sides with respect to x, we obtain
`(1)/(y)cdot(dy)/(dx)=log (log x) (d)/(dx)(cos x) + cos xcdot(d)/(dx)[log (log x)]`
`"or "(1)/(y)cdot(dy)/(dx)=-sin x log (log x) + cos xcdot(1)/(log x)cdot(d)/(dx)(log x)`
`"or "(dy)/(dx)=y[-sin xlog (log x)+ (cos x)/(log x)cdot(1)/(x)]`
`=(log x)^(cos x)[(cos x)/( x log x)-sin x log (log x)]`
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