Question

If f(x) = `cos^-1[(1 - (logx)^2)/(1 + (logx)^2)]`, then f'(e) = ______.

Options

• `1/e`

• `2/e^2`

• `2/e`

• 1

Answer 1

If f(x) = `cos^-1[(1 - (logx)^2)/(1 + (logx)^2)]`, then f'(e) = `underlinebb(1/e)`.

Explanation:

f(x) = `cos^-1[(1 - (logx)^2)/(1 + (logx)^2)]`

Let 1 + (log x)² = u

`\implies` 1 – (log x)² = 2 – u

 `\implies` f'(u) = `cos^-1((2 - u)/u) = cos^-1(2/u - 1)`

 `\implies` f'(u) = `(((2/u^2))/sqrt(1 - (2/u - 1)^2)) = 1/(usqrt(u - 1)`

 `\implies` f'(u) = `1/((1 + ((log x)^2)sqrt((logx)^2`

= `1/(logx(1 + (logx)^2)`

 `\implies` f'(e) = `1/(loge(1 + (log e)^2)) = 1/2`