If f(x) = xtan^-1 x, what is f'(√(3))?

Question 41

MediumOriginal question

If $f(x) = x\tan^{-1} x$ , what is $f'\left(\sqrt{3}\right)$ ?

\frac{\sqrt{3}}{4}

Correct answerYour answer

\frac{\pi}{6} + \frac{\sqrt{3}}{4}

Correct answerYour answer

\frac{\pi}{3}

Correct answerYour answer

\frac{\pi}{3} + \frac{\sqrt{3}}{4}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Product rule: $f'(x) = \tan^{-1} x + \frac{x}{1 + x^{2}}$ .
At $\sqrt{3}$ : $\tan^{-1}\sqrt{3} = \frac{\pi}{3}$ and $\frac{\sqrt{3}}{1 + 3} = \frac{\sqrt{3}}{4}$ , so $f' = \frac{\pi}{3} + \frac{\sqrt{3}}{4}$ .

Worth noting

$\frac{\pi}{3}$ keeps only the term $\tan^{-1} x$ and drops $x \cdot \frac{1}{1 + x^{2}}$ .

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