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Derivative of 1xarctan⁡(x)\frac{\frac{1}{x}}{\arctan\left(x\right)}

The derivative of 1xarctan⁡(x)\frac{\frac{1}{x}}{\arctan\left(x\right)} is −1x2arctan⁡(x)−11+x2xarctan⁡(x)2\frac{\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}}{\arctan\left(x\right)^{2}}.

f(x) =
ddx[1xarctan⁡(x)]=−1x2arctan⁡(x)−11+x2xarctan⁡(x)2\frac{d}{dx}\left[\frac{\frac{1}{x}}{\arctan\left(x\right)}\right] = \frac{\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}}{\arctan\left(x\right)^{2}}

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Graph of f(x) = 1 / x / atan(x) and its derivative-3-2-1123-2020406080slope -2.084f(x)f'(x)tangent at x = 1
Graph of f(x) = 1 / x / atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -2.0838.

How to work it out

  1. 1

    We differentiate f(x)=1xarctan⁡(x)f(x) = \frac{\frac{1}{x}}{\arctan\left(x\right)} with respect to x. The rule that applies is the quotient rule.

    f(x)=1xarctan⁡(x)f(x) = \frac{\frac{1}{x}}{\arctan\left(x\right)}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=1xu = \frac{1}{x} and v=arctan⁡(x)v = \arctan\left(x\right).

    u=1x,v=arctan⁡(x)u = \frac{1}{x},\quad v = \arctan\left(x\right)
  3. 3

    Differentiate each piece:

    u′=−1x2,v′=11+x2u' = \frac{-1}{x^{2}},\quad v' = \frac{1}{1 + x^{2}}
  4. 4

    Put them in the rule:

    f′(x)=(−1x2)(arctan⁡(x))−(1x)(11+x2)(arctan⁡(x))2f'(x) = \frac{\left(\frac{-1}{x^{2}}\right)\left(\arctan\left(x\right)\right) - \left(\frac{1}{x}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}}
  5. 5

    Simplify:

    f′(x)=−1x2arctan⁡(x)−11+x2xarctan⁡(x)2f'(x) = \frac{\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}}{\arctan\left(x\right)^{2}}
  6. 6

    Differentiating once more gives the second derivative, which tells you about curvature:

    f′′(x)=(−((−2)x)(x2)2arctan⁡(x)+−1x211+x2−−(2x)(1+x2)2x−11+x2x2)arctan⁡(x)2−(−1x2arctan⁡(x)−11+x2x)2arctan⁡(x)11+x2(arctan⁡(x)2)2f''(x) = \frac{\left(\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}} \arctan\left(x\right) + \frac{-1}{x^{2}} \frac{1}{1 + x^{2}} - \frac{\frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}} x - \frac{1}{1 + x^{2}}}{x^{2}}\right) \arctan\left(x\right)^{2} - \left(\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\frac{1}{x}}{\arctan\left(x\right)}\right] = \frac{\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}}{\arctan\left(x\right)^{2}} \]
\[ f(x) = \frac{\frac{1}{x}}{\arctan\left(x\right)} \]
\[ u = \frac{1}{x},\quad v = \arctan\left(x\right) \]
\[ u' = \frac{-1}{x^{2}},\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \frac{\left(\frac{-1}{x^{2}}\right)\left(\arctan\left(x\right)\right) - \left(\frac{1}{x}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}}{\arctan\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}} \arctan\left(x\right) + \frac{-1}{x^{2}} \frac{1}{1 + x^{2}} - \frac{\frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}} x - \frac{1}{1 + x^{2}}}{x^{2}}\right) \arctan\left(x\right)^{2} - \left(\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}} \]

Values of f and f' for 1 / x / atan(x)

xf(x)f'(x)
-20.451610510.30738608
-11.27323952.083809
0.54.3136209-16.070172
11.2732395-2.083809
20.45161051-0.30738608
30.26687039-0.11032274

Questions

What is the derivative of 1 / x / atan(x)?

The derivative of 1xarctan⁡(x)\frac{\frac{1}{x}}{\arctan\left(x\right)} is −1x2arctan⁡(x)−11+x2xarctan⁡(x)2\frac{\frac{-1}{x^{2}} \arctan\left(x\right) - \frac{\frac{1}{1 + x^{2}}}{x}}{\arctan\left(x\right)^{2}}.

What is the second derivative of 1 / x / atan(x)?

((-(-2x) / (x2x^{2})^2 * atan(x) + -1 / x2x^{2} * 1 / (1 + x2x^{2}) - (-(2x) / (1 + x2x^{2})^2 * x - 1 / (1 + x2x^{2})) / x2x^{2}) * atan(x)^2 - (-1 / x2x^{2} * atan(x) - 1 / (1 + x2x^{2}) / x) * 2atan(x) * 1 / (1 + x2x^{2})) / (atan(x)^2)^2.

Which rule do you use for 1 / x / atan(x)?

The quotient rule.

What is the slope of 1 / x / atan(x) at x = 1?

-2.083809.

Related derivatives

    Derivative of 1 / x / atan(x): (-1 / x^2 * atan(x) - 1 / (1 + x^2) / x) / atan(x)^2 (with steps) | WordGPT