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

The derivative of arctan⁡(x)1x\frac{\arctan\left(x\right)}{\frac{1}{x}} is 1x1+x2−arctan⁡(x)−1x21x2\frac{\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

f(x) =
ddx[arctan⁡(x)1x]=1x1+x2−arctan⁡(x)−1x21x2\frac{d}{dx}\left[\frac{\arctan\left(x\right)}{\frac{1}{x}}\right] = \frac{\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}

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Graph of f(x) = atan(x) / (1 / x) and its derivative-3-2-1123-4-3-2-11234slope 1.285f(x)f'(x)tangent at x = 1
Graph of f(x) = atan(x) / (1 / x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.2854.

How to work it out

  1. 1

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

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

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

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=1x1+x2−arctan⁡(x)−1x21x2f'(x) = \frac{\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}
  6. 6

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

    f′′(x)=(−1x2(1+x2)−2xx(1+x2)2−(−1x21+x2+arctan⁡(x)−((−2)x)(x2)2))1x2−(1x1+x2−arctan⁡(x)−1x2)21x−1x2(1x2)2f''(x) = \frac{\left(\frac{\frac{-1}{x^{2}} \left(1 + x^{2}\right) - \frac{2 x}{x}}{\left(1 + x^{2}\right)^{2}} - \left(\frac{\frac{-1}{x^{2}}}{1 + x^{2}} + \arctan\left(x\right) \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\arctan\left(x\right)}{\frac{1}{x}}\right] = \frac{\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f(x) = \frac{\arctan\left(x\right)}{\frac{1}{x}} \]
\[ u = \arctan\left(x\right),\quad v = \frac{1}{x} \]
\[ u' = \frac{1}{1 + x^{2}},\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(\frac{1}{x}\right) - \left(\arctan\left(x\right)\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}} \]
\[ f'(x) = \frac{\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f''(x) = \frac{\left(\frac{\frac{-1}{x^{2}} \left(1 + x^{2}\right) - \frac{2 x}{x}}{\left(1 + x^{2}\right)^{2}} - \left(\frac{\frac{-1}{x^{2}}}{1 + x^{2}} + \arctan\left(x\right) \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}} \]

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

xf(x)f'(x)
-22.2142974-1.5071487
-10.78539816-1.2853982
0.50.23182380.86364761
10.785398161.2853982
22.21429741.5071487
33.74713731.5490458

Questions

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

The derivative of arctan⁡(x)1x\frac{\arctan\left(x\right)}{\frac{1}{x}} is 1x1+x2−arctan⁡(x)−1x21x2\frac{\frac{\frac{1}{x}}{1 + x^{2}} - \arctan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

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

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

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

The quotient rule.

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

1.2853982.

Related derivatives

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