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Question
Choose the correct option from the given alternatives :
If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?
Options
`(2.4^x.log4)/(1 + 4^(2x)`
`(4^(x + 1).log2)/(1 + 4^(2x)`
`(4^(x + 1).log4)/(1 + 4^(4x)`
`(2^(2^((x + 1)).log2))/(1 + 2^(4x)`
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Solution
`(4^(x + 1)log4)/(1 + 4^(4x))`
[Hint : Put 4x = tanθ. Thenθ = tan–1(4x)
∴ f(x) = `sin^-1((2tanθ)/(1 + tan^2θ))`
= sin–1(sin2θ)
= 2θ
= 2tan–1(4x)
∴ f'(x) = `2 xx (1)/(1 + (4^x)^2) xx 4^xlog4`
= `(2.4^x.log4)/(1 + 4^(2x)` ... (a)
= `(2.4^x.2log2)/(1 + 4^(2x)`
= `(4^(x + 1).log2)/(1 + 4^(2x)` ...(b)
= `((2^2)^(x + 1).log2)/(1 + 2^(4x)`
= `(2^(2^((x + 1)).log2))/(1 + 2^(4x)` ...(d)]
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