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Derivative of arctan⁡(x)x2\frac{\arctan\left(x\right)}{x^{2}}

The derivative of arctan⁡(x)x2\frac{\arctan\left(x\right)}{x^{2}} is x21+x2−arctan⁡(x)2x(x2)2\frac{\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x}{\left(x^{2}\right)^{2}}.

f(x) =
ddx[arctan⁡(x)x2]=x21+x2−arctan⁡(x)2x(x2)2\frac{d}{dx}\left[\frac{\arctan\left(x\right)}{x^{2}}\right] = \frac{\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x}{\left(x^{2}\right)^{2}}

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Graph of f(x) = atan(x) / x^2 and its derivative-3-2-1123-15-10-55slope -1.071f(x)f'(x)tangent at x = 1
Graph of f(x) = atan(x) / x^2 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.0708.

How to work it out

  1. 1

    We differentiate f(x)=arctan⁡(x)x2f(x) = \frac{\arctan\left(x\right)}{x^{2}} with respect to x. The rule that applies is the quotient rule.

    f(x)=arctan⁡(x)x2f(x) = \frac{\arctan\left(x\right)}{x^{2}}
  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=x2v = x^{2}.

    u=arctan⁡(x),v=x2u = \arctan\left(x\right),\quad v = x^{2}
  3. 3

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=x21+x2−arctan⁡(x)2x(x2)2f'(x) = \frac{\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x}{\left(x^{2}\right)^{2}}
  6. 6

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

    f′′(x)=(2x(1+x2)−x22x(1+x2)2−(2x1+x2+2arctan⁡(x)))(x2)2−(x21+x2−arctan⁡(x)2x)2x22x((x2)2)2f''(x) = \frac{\left(\frac{2 x \left(1 + x^{2}\right) - x^{2} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{2 x}{1 + x^{2}} + 2 \arctan\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\arctan\left(x\right)}{x^{2}}\right] = \frac{\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f(x) = \frac{\arctan\left(x\right)}{x^{2}} \]
\[ u = \arctan\left(x\right),\quad v = x^{2} \]
\[ u' = \frac{1}{1 + x^{2}},\quad v' = 2 x \]
\[ f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(x^{2}\right) - \left(\arctan\left(x\right)\right)\left(2 x\right)}{\left(x^{2}\right)^{2}} \]
\[ f'(x) = \frac{\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{2 x \left(1 + x^{2}\right) - x^{2} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{2 x}{1 + x^{2}} + 2 \arctan\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}} \]

Values of f and f' for atan(x) / x2x^{2}

xf(x)f'(x)
-2-0.27678718-0.22678718
-1-0.78539816-1.0707963
0.51.8545904-4.2183617
10.78539816-1.0707963
20.27678718-0.22678718
30.13878286-0.081410798

Questions

What is the derivative of atan(x) / x2x^{2}?

The derivative of arctan⁡(x)x2\frac{\arctan\left(x\right)}{x^{2}} is x21+x2−arctan⁡(x)2x(x2)2\frac{\frac{x^{2}}{1 + x^{2}} - \arctan\left(x\right) 2 x}{\left(x^{2}\right)^{2}}.

What is the second derivative of atan(x) / x2x^{2}?

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

Which rule do you use for atan(x) / x2x^{2}?

The quotient rule.

What is the slope of atan(x) / x2x^{2} at x = 1?

-1.0707963.

Related derivatives

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