Question

If f(x) = x tan^–1 x, then f'(1) is equal to ______.

Options

• `π/4 - 1/2`

• `π/4 + 1/2`

• `-π/4 - 1/2`

• `-π/4 + 1/2`

Answer 1

If f(x) = x tan^–1 x, then f'(1) is equal to `underlinebb(π/4 + 1/2)`.

Explanation:

Given f(x) = x tan^–1 x

Now, f'(x) = (x tan^–1 x)'

Using product rule,

= (x)' tan^–1 x + x (tan^–1 x)'

= `1 xx tan^-1x + x xx 1/(1 + x^2)`

= `tan^-1 x + x/(1 + x^2)`

Now, f'(1) = `tan^-1 (1) + 1/(1 + 1^2)`

= `tan^-1 (1) + 1/(1 + 1)`

= `tan^-1 (1) + 1/2`

= `π/4 + 1/2`