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Question
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
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Solution
`y = sin^(-1) [(2.2^x)/(1 +(2^x)^2)]`
put 2x = tan θ
`∴ y = sin^(-1) [(2 tan theta ) /(1 + tan^2 theta)]`
= sin-1 [ sin 2θ ]
= 2θ
y = 2 tan-1 ( 2x )
Differentiating wrt x,
`(dy)/(dx) = 2/(1 +(2^x) )xx d/(dx) (2^x)`
`= 2/(1 + (2^x)^2) xx 2^x log 2 = (2 ^ (x+ 1))/(1 + 4^x) log 2 =" sin y log" 2`
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