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Question
Differentiate the following w.r.t. x : (sin x)x
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Solution
Let y = (sin x)x
Then log y = log(sin x)x = x.log(sin x)
Differentiating both sides w.r.t. x, we get
`(1)/y."dy"/"dx" = "d"/"dx"[x.log(sinx)]`
= `x."d"/"dx"[log(sinx)] + log(sinx)."d"/"dx"(x)`
= `x xx (1)/sinx."d"/"dx"(sinx) + log(sinx) xx 1`
∴ `"dy"/"dx" = y[x xx 1/sinx. cosx + log(sinx)]`
= (sin x)x[x cotx + log (sinx)].
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