AP Calculus AB and BC

Derivative of x/(x^2+1): Quotient Rule

The derivative of x over x squared plus 1 is 1 minus x squared, all over x squared plus 1 squared. The quotient rule gives it directly, and the numerator factors as 1 minus x times 1 plus x, so the derivative is zero at x equals 1 and negative 1.

ddx[xx2+1]=1−x2(x2+1)2\frac{d}{dx}\left[\frac{x}{x^{2}+1}\right] = \frac{1-x^{2}}{\left(x^{2}+1\right)^{2}}

Applying the quotient rule

ddxuv=u′v−uv′v2\frac{d}{dx}\frac{u}{v} = \frac{u'v - uv'}{v^{2}}

With u=xu = x and v=x2+1v = x^{2}+1, so u′=1u' = 1 and v′=2xv' = 2x.

(1)(x2+1)−(x)(2x)(x2+1)2=x2+1−2x2(x2+1)2=1−x2(x2+1)2\frac{(1)\left(x^{2}+1\right) - (x)(2x)}{\left(x^{2}+1\right)^{2}} = \frac{x^{2}+1-2x^{2}}{\left(x^{2}+1\right)^{2}} = \frac{1-x^{2}}{\left(x^{2}+1\right)^{2}}

What the sign tells you

The denominator is always positive, so the sign of the derivative is the sign of 1−x21-x^{2}. That is positive on −1<x<1-1 < x < 1 and negative outside, so the function rises between the two critical points and falls beyond them.

By the First Derivative Test there is a minimum at x=−1x = -1 and a maximum at x=1x = 1, with values −12-\frac{1}{2} and 12\frac{1}{2}.

Common mistakes

  • Forgetting to subtract in the numerator. The order is u′v−uv′u'v - uv', and swapping it flips every sign.
  • Squaring only part of the denominator. The whole of x2+1x^{2}+1 gets squared.
  • Expanding (x2+1)2\left(x^{2}+1\right)^{2} before finding critical points. Leaving it factored makes the sign analysis immediate.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

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

It is 1−x2(x2+1)2\frac{1-x^{2}}{\left(x^{2}+1\right)^{2}}.

Where are the critical points?

At x=±1x = \pm 1, where the numerator 1−x21-x^{2} is zero. The denominator never vanishes.

Which is the maximum?

x=1x = 1, where the derivative changes from positive to negative. The maximum value is 12\frac{1}{2}.