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Question

Find the derivative with respect to x of the function (logcosxsinx)(logsinxcosx)1+sin12x1+x2 at x=π4

A
8(4π2+161log2)
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B
8(4(π+4)21log2)
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C
8(4π2+16+1log2)
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D
8(4(π+4)21log2)
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Solution

The correct option is A 8(4π2+161log2)
Let y=(logcosxsinx)(logsinxcosx)1+sin12x1+x2

=(logcosxsinx)2+2tan1x[logba=1logab] =(logesinxlogecosx)2+2tan1x.

dydx=2(logsinxlogcosx)cotx.logcosx+tanxlogsinx(logcosx)2+21+x2

Hence at x=π4.

we have dydx=2.log(1/2)log(1/2)1.log(1/2)+1.log(1/2)[log(1/2)]2+21+(π2/16)=8log2+32π2+16

=8(4π2+161log2).

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