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Derivative of arctan⁡(x)x5\frac{\arctan\left(x\right)}{x^{5}}

The derivative of arctan⁡(x)x5\frac{\arctan\left(x\right)}{x^{5}} is x51+x2−arctan⁡(x)5x4(x5)2\frac{\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}}{\left(x^{5}\right)^{2}}.

f(x) =
ddx[arctan⁡(x)x5]=x51+x2−arctan⁡(x)5x4(x5)2\frac{d}{dx}\left[\frac{\arctan\left(x\right)}{x^{5}}\right] = \frac{\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}}{\left(x^{5}\right)^{2}}

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Graph of f(x) = atan(x) / x^5 and its derivative-3-2-112350010001500200025003000slope -3.427f(x)f'(x)tangent at x = 1
Graph of f(x) = atan(x) / x^5 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -3.427.

How to work it out

  1. 1

    We differentiate f(x)=arctan⁡(x)x5f(x) = \frac{\arctan\left(x\right)}{x^{5}} with respect to x. The rule that applies is the quotient rule.

    f(x)=arctan⁡(x)x5f(x) = \frac{\arctan\left(x\right)}{x^{5}}
  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=x5v = x^{5}.

    u=arctan⁡(x),v=x5u = \arctan\left(x\right),\quad v = x^{5}
  3. 3

    Differentiate each piece:

    u′=11+x2,v′=5x4u' = \frac{1}{1 + x^{2}},\quad v' = 5 x^{4}
  4. 4

    Put them in the rule:

    f′(x)=(11+x2)(x5)−(arctan⁡(x))(5x4)(x5)2f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(x^{5}\right) - \left(\arctan\left(x\right)\right)\left(5 x^{4}\right)}{\left(x^{5}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=x51+x2−arctan⁡(x)5x4(x5)2f'(x) = \frac{\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}}{\left(x^{5}\right)^{2}}
  6. 6

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

    f′′(x)=(5x4(1+x2)−x52x(1+x2)2−(5x41+x2+arctan⁡(x)20x3))(x5)2−(x51+x2−arctan⁡(x)5x4)2x55x4((x5)2)2f''(x) = \frac{\left(\frac{5 x^{4} \left(1 + x^{2}\right) - x^{5} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{5 x^{4}}{1 + x^{2}} + \arctan\left(x\right) 20 x^{3}\right)\right) \left(x^{5}\right)^{2} - \left(\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}\right) 2 x^{5} 5 x^{4}}{\left(\left(x^{5}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\arctan\left(x\right)}{x^{5}}\right] = \frac{\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}}{\left(x^{5}\right)^{2}} \]
\[ f(x) = \frac{\arctan\left(x\right)}{x^{5}} \]
\[ u = \arctan\left(x\right),\quad v = x^{5} \]
\[ u' = \frac{1}{1 + x^{2}},\quad v' = 5 x^{4} \]
\[ f'(x) = \frac{\left(\frac{1}{1 + x^{2}}\right)\left(x^{5}\right) - \left(\arctan\left(x\right)\right)\left(5 x^{4}\right)}{\left(x^{5}\right)^{2}} \]
\[ f'(x) = \frac{\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}}{\left(x^{5}\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{5 x^{4} \left(1 + x^{2}\right) - x^{5} 2 x}{\left(1 + x^{2}\right)^{2}} - \left(\frac{5 x^{4}}{1 + x^{2}} + \arctan\left(x\right) 20 x^{3}\right)\right) \left(x^{5}\right)^{2} - \left(\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}\right) 2 x^{5} 5 x^{4}}{\left(\left(x^{5}\right)^{2}\right)^{2}} \]

Values of f and f' for atan(x) / x5x^{5}

xf(x)f'(x)
-20.0345983970.080245994
-10.785398163.4269908
0.514.836723-122.76723
10.78539816-3.4269908
20.034598397-0.080245994
30.0051401061-0.0081553208

Questions

What is the derivative of atan(x) / x5x^{5}?

The derivative of arctan⁡(x)x5\frac{\arctan\left(x\right)}{x^{5}} is x51+x2−arctan⁡(x)5x4(x5)2\frac{\frac{x^{5}}{1 + x^{2}} - \arctan\left(x\right) 5 x^{4}}{\left(x^{5}\right)^{2}}.

What is the second derivative of atan(x) / x5x^{5}?

(((5 * x4x^{4} * (1 + x2x^{2}) - x5x^{5} * 2x) / (1 + x2x^{2})^2 - (5 * x4x^{4} / (1 + x2x^{2}) + atan(x) * 20 * x3x^{3})) * (x5x^{5})^2 - (x5x^{5} / (1 + x2x^{2}) - atan(x) * 5 * x4x^{4}) * 2 * x5x^{5} * 5 * x4x^{4}) / ((x5x^{5})^2)^2.

Which rule do you use for atan(x) / x5x^{5}?

The quotient rule.

What is the slope of atan(x) / x5x^{5} at x = 1?

-3.4269908.

Related derivatives

    Derivative of atan(x) / x^5: (x^5 / (1 + x^2) - atan(x) * 5 * x^4) / (x^5)^2 (with steps) | WordGPT