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Question
Differentiate the function with respect to x.
`(log x)^(cos x)`
Solution
Let, y = `(log x)^(cos x)`
Taking logarithm of both sides,
log y = log (log x)cos x
= cos x log (log x), ...[∵ log mn = n log m]
Differentiating both sides with respect to x,
`1/y dy/dx = cos x d/dx log (log x) + log (log x) d/dx cos x`
`= cos x * 1/(log x) d/dx (log x) + log (log x) (- sin x)`
`= cos x * 1/(log x) * 1/x - sin x log (log x)`
`= - sin x log (log x) + (cos x)/(x log x)`
`therefore dy/dx = y [- sin x log (log x) + (cos x)/(x log x)]`
`= (log x)^(cos x) [- sin x log (log x) + (cos x)/(x log x)]`
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