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Question

If the first derivative of the function f(x)=sin(cos1(122x1+22x)) with respect to x at x=1 is balog2, where a,b are coprime, then |4b2a|=

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Solution

Let t=2x
=sin(cos1(1t21+t2))
=sin(sin1(2t1+t2))

=2t1+t2

=2.2x1+22x
Differentiating
f(x)=(1+22x)(2.2xlog2)2.2x.22xlog2.2(1+22x)2

f(1)=5.2.2log22.2.4.2log2(5)2

f(1)=balog2=1225log2

a=25 b=12

|4b2a|=|4850|=|2|=2


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