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Question
Differentiate the following with respect to x.
xsin x
Solution
Let y = xsin x
Taking logarithm on both sides we get,
log y = log(xsin x)
log y = sin x log x
Differentiating with respect to x,
`1/y * "dy"/"dx" = sin x "d"/"dx" (log x) + log x "d"/"dx" (sin x)`
`1/y * "dy"/"dx" = sin x (1/x) + log x (cos x)`
`"dy"/"dx" = y[(sin x)/x + cos x log x]`
`therefore "dy"/"dx" = x^(sin x) [(sin x)/x + cos x log x]`
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