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Question

Iff(x)=(logcostanx)(logtanxcotx)-1+tan-1x4-x2 then f'(1) is


A

0

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B

-2

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C

13

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D

3

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Solution

The correct option is C

13


Explanation for correct option:

Step 1: Simplifying the given function:

f(x)=logtanxlogcotxlogcotxlogtanx-1+tan-1x4-x2[logab=logbloga]=logtanx-logtanx.logtanx-logtanx+tan-1x4-x2[log(cotx)=log(1tanx)=-logtanx]=1+tan-12sinθ4-4sin2θputtingx=2sinθ=1+tan-12sinθ2cosθ=1+tan-1(tanθ)=1+θ=1+sin-1x2x=2sinθ

Step 2: Find the value of f'(1):

Differentiated the simplified function with respect to x.

f'(x)=11-x24.12=14-x2

Substitute x=1 in f'(x), we get

f'(1)=14-1=13

Hence, the correct option is C.


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