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)`

Answer 1

`(4^(x + 1)log4)/(1 + 4^(4x))`

[Hint : Put 4^x = tanθ. Thenθ = tan^–1(4^x)

∴ f(x) = `sin^-1((2tanθ)/(1 + tan^2θ))`

= sin^–1(sin2θ)

= 2θ

= 2tan^–1(4^x)

∴ 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)]