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Derivative of x5arctan⁡(x)\frac{x^{5}}{\arctan\left(x\right)}

The derivative of x5arctan⁡(x)\frac{x^{5}}{\arctan\left(x\right)} is 5x4arctan⁡(x)−x511+x2arctan⁡(x)2\frac{5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

f(x) =
ddx[x5arctan⁡(x)]=5x4arctan⁡(x)−x511+x2arctan⁡(x)2\frac{d}{dx}\left[\frac{x^{5}}{\arctan\left(x\right)}\right] = \frac{5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}

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Graph of f(x) = x^5 / atan(x) and its derivative-3-2-1123-150-100-5050100150200slope 5.556f(x)f'(x)tangent at x = 1
Graph of f(x) = x^5 / atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 5.5556.

How to work it out

  1. 1

    We differentiate f(x)=x5arctan⁡(x)f(x) = \frac{x^{5}}{\arctan\left(x\right)} with respect to x. The rule that applies is the quotient rule.

    f(x)=x5arctan⁡(x)f(x) = \frac{x^{5}}{\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=x5u = x^{5} and v=arctan⁡(x)v = \arctan\left(x\right).

    u=x5,v=arctan⁡(x)u = x^{5},\quad v = \arctan\left(x\right)
  3. 3

    Differentiate each piece:

    u′=5x4,v′=11+x2u' = 5 x^{4},\quad v' = \frac{1}{1 + x^{2}}
  4. 4

    Put them in the rule:

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

    Simplify:

    f′(x)=5x4arctan⁡(x)−x511+x2arctan⁡(x)2f'(x) = \frac{5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}
  6. 6

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

    f′′(x)=(20x3arctan⁡(x)+5x411+x2−(5x411+x2+x5−(2x)(1+x2)2))arctan⁡(x)2−(5x4arctan⁡(x)−x511+x2)2arctan⁡(x)11+x2(arctan⁡(x)2)2f''(x) = \frac{\left(20 x^{3} \arctan\left(x\right) + 5 x^{4} \frac{1}{1 + x^{2}} - \left(5 x^{4} \frac{1}{1 + x^{2}} + x^{5} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}\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
\[ \frac{d}{dx}\left[\frac{x^{5}}{\arctan\left(x\right)}\right] = \frac{5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f(x) = \frac{x^{5}}{\arctan\left(x\right)} \]
\[ u = x^{5},\quad v = \arctan\left(x\right) \]
\[ u' = 5 x^{4},\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \frac{\left(5 x^{4}\right)\left(\arctan\left(x\right)\right) - \left(x^{5}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(20 x^{3} \arctan\left(x\right) + 5 x^{4} \frac{1}{1 + x^{2}} - \left(5 x^{4} \frac{1}{1 + x^{2}} + x^{5} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-228.903073-67.036509
-11.2732395-5.5556283
0.50.0674003260.55770748
11.27323955.5556283
228.90307367.036509
3194.54851308.67175

Questions

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

The derivative of x5arctan⁡(x)\frac{x^{5}}{\arctan\left(x\right)} is 5x4arctan⁡(x)−x511+x2arctan⁡(x)2\frac{5 x^{4} \arctan\left(x\right) - x^{5} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

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

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

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

The quotient rule.

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

5.5556283.

Related derivatives

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