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Derivative of arctan⁡(x)x3\frac{\arctan\left(x\right)}{x^{3}}

The derivative of arctan⁡(x)x3\frac{\arctan\left(x\right)}{x^{3}} is x31+x2−arctan⁡(x)3x2(x3)2\frac{\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}.

f(x) =
ddx[arctan⁡(x)x3]=x31+x2−arctan⁡(x)3x2(x3)2\frac{d}{dx}\left[\frac{\arctan\left(x\right)}{x^{3}}\right] = \frac{\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}

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Graph of f(x) = atan(x) / x^3 and its derivative-3-2-1123-2020406080slope -1.856f(x)f'(x)tangent at x = 1
Graph of f(x) = atan(x) / x^3 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.8562.

How to work it out

  1. 1

    We differentiate f(x)=arctan⁡(x)x3f(x) = \frac{\arctan\left(x\right)}{x^{3}} with respect to x. The rule that applies is the quotient rule.

    f(x)=arctan⁡(x)x3f(x) = \frac{\arctan\left(x\right)}{x^{3}}
  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=x3v = x^{3}.

    u=arctan⁡(x),v=x3u = \arctan\left(x\right),\quad v = x^{3}
  3. 3

    Differentiate each piece:

    u′=11+x2,v′=3x2u' = \frac{1}{1 + x^{2}},\quad v' = 3 x^{2}
  4. 4

    Put them in the rule:

    f′(x)=(11+x2)(x3)−(arctan⁡(x))(3x2)(x3)2f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(x^{3}\right) - \left(\arctan\left(x\right)\right)\left(3 x^{2}\right)}{\left(x^{3}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=x31+x2−arctan⁡(x)3x2(x3)2f'(x) = \frac{\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}
  6. 6

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

    f′′(x)=(3x2(1+x2)−x32x(1+x2)2−(3x21+x2+arctan⁡(x)6x))(x3)2−(x31+x2−arctan⁡(x)3x2)2x33x2((x3)2)2f''(x) = \frac{\left(\frac{3 x^{2} \left(1 + x^{2}\right) - x^{3} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{3 x^{2}}{1 + x^{2}} + \arctan\left(x\right) 6 x\right)\right) \left(x^{3}\right)^{2} - \left(\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}\right) 2 x^{3} 3 x^{2}}{\left(\left(x^{3}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\arctan\left(x\right)}{x^{3}}\right] = \frac{\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}} \]
\[ f(x) = \frac{\arctan\left(x\right)}{x^{3}} \]
\[ u = \arctan\left(x\right),\quad v = x^{3} \]
\[ u' = \frac{1}{1 + x^{2}},\quad v' = 3 x^{2} \]
\[ f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(x^{3}\right) - \left(\arctan\left(x\right)\right)\left(3 x^{2}\right)}{\left(x^{3}\right)^{2}} \]
\[ f'(x) = \frac{\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{3 x^{2} \left(1 + x^{2}\right) - x^{3} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{3 x^{2}}{1 + x^{2}} + \arctan\left(x\right) 6 x\right)\right) \left(x^{3}\right)^{2} - \left(\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}\right) 2 x^{3} 3 x^{2}}{\left(\left(x^{3}\right)^{2}\right)^{2}} \]

Values of f and f' for atan(x) / x3x^{3}

xf(x)f'(x)
-20.138393590.18259038
-10.785398161.8561945
0.53.7091809-15.855085
10.78539816-1.8561945
20.13839359-0.18259038
30.046260955-0.042557251

Questions

What is the derivative of atan(x) / x3x^{3}?

The derivative of arctan⁡(x)x3\frac{\arctan\left(x\right)}{x^{3}} is x31+x2−arctan⁡(x)3x2(x3)2\frac{\frac{x^{3}}{1 + x^{2}} - \arctan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}.

What is the second derivative of atan(x) / x3x^{3}?

(((3 * x2x^{2} * (1 + x2x^{2}) - x3x^{3} * 2x) / (1 + x2x^{2})^2 - (3 * x2x^{2} / (1 + x2x^{2}) + atan(x) * 6x)) * (x3x^{3})^2 - (x3x^{3} / (1 + x2x^{2}) - atan(x) * 3 * x2x^{2}) * 2 * x3x^{3} * 3 * x2x^{2}) / ((x3x^{3})^2)^2.

Which rule do you use for atan(x) / x3x^{3}?

The quotient rule.

What is the slope of atan(x) / x3x^{3} at x = 1?

-1.8561945.

Related derivatives

    Derivative of atan(x) / x^3: (x^3 / (1 + x^2) - atan(x) * 3 * x^2) / (x^3)^2 (with steps) | WordGPT