What is the derivative of x/(x^2 + 1)?

Question 28

HardOriginal question

What is the derivative of $\frac{x}{x^{2} + 1}$ ?

\frac{x^{2} - 1}{\left(x^{2} + 1\right)^{2}}

Correct answerYour answer

\frac{1}{2 x}

Correct answerYour answer

\frac{3 x^{2} + 1}{\left(x^{2} + 1\right)^{2}}

Correct answerYour answer

\frac{1 - x^{2}}{\left(x^{2} + 1\right)^{2}}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ , with $u = x$ and $v = x^{2} + 1$ .
This simplifies to $\frac{1 - x^{2}}{\left(x^{2} + 1\right)^{2}}$ .

Worth noting

$\frac{x^{2} - 1}{\left(x^{2} + 1\right)^{2}}$ has the numerator the wrong way round: it is $u'v - uv'$ , not $uv' - u'v$ .

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