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Derivative of x4arctan⁡(x)\frac{x^{4}}{\arctan\left(x\right)}

The derivative of x4arctan⁡(x)\frac{x^{4}}{\arctan\left(x\right)} is 4x3arctan⁡(x)−x411+x2arctan⁡(x)2\frac{4 x^{3} \arctan\left(x\right) - x^{4} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

f(x) =
ddx[x4arctan⁡(x)]=4x3arctan⁡(x)−x411+x2arctan⁡(x)2\frac{d}{dx}\left[\frac{x^{4}}{\arctan\left(x\right)}\right] = \frac{4 x^{3} \arctan\left(x\right) - x^{4} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}

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Graph of f(x) = x^4 / atan(x) and its derivative-3-2-1123-40-20204060slope 4.282f(x)f'(x)tangent at x = 1
Graph of f(x) = x^4 / atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 4.2824.

How to work it out

  1. 1

    We differentiate f(x)=x4arctan⁡(x)f(x) = \frac{x^{4}}{\arctan\left(x\right)} with respect to x. The rule that applies is the quotient rule.

    f(x)=x4arctan⁡(x)f(x) = \frac{x^{4}}{\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=x4u = x^{4} and v=arctan⁡(x)v = \arctan\left(x\right).

    u=x4,v=arctan⁡(x)u = x^{4},\quad v = \arctan\left(x\right)
  3. 3

    Differentiate each piece:

    u′=4x3,v′=11+x2u' = 4 x^{3},\quad v' = \frac{1}{1 + x^{2}}
  4. 4

    Put them in the rule:

    f′(x)=(4x3)(arctan⁡(x))−(x4)(11+x2)(arctan⁡(x))2f'(x) = \frac{\left(4 x^{3}\right)\left(\arctan\left(x\right)\right) - \left(x^{4}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}}
  5. 5

    Simplify:

    f′(x)=4x3arctan⁡(x)−x411+x2arctan⁡(x)2f'(x) = \frac{4 x^{3} \arctan\left(x\right) - x^{4} \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)=(12x2arctan⁡(x)+4x311+x2−(4x311+x2+x4−(2x)(1+x2)2))arctan⁡(x)2−(4x3arctan⁡(x)−x411+x2)2arctan⁡(x)11+x2(arctan⁡(x)2)2f''(x) = \frac{\left(12 x^{2} \arctan\left(x\right) + 4 x^{3} \frac{1}{1 + x^{2}} - \left(4 x^{3} \frac{1}{1 + x^{2}} + x^{4} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(4 x^{3} \arctan\left(x\right) - x^{4} \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
\[ \frac{d}{dx}\left[\frac{x^{4}}{\arctan\left(x\right)}\right] = \frac{4 x^{3} \arctan\left(x\right) - x^{4} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f(x) = \frac{x^{4}}{\arctan\left(x\right)} \]
\[ u = x^{4},\quad v = \arctan\left(x\right) \]
\[ u' = 4 x^{3},\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \frac{\left(4 x^{3}\right)\left(\arctan\left(x\right)\right) - \left(x^{4}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{4 x^{3} \arctan\left(x\right) - x^{4} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(12 x^{2} \arctan\left(x\right) + 4 x^{3} \frac{1}{1 + x^{2}} - \left(4 x^{3} \frac{1}{1 + x^{2}} + x^{4} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(4 x^{3} \arctan\left(x\right) - x^{4} \frac{1}{1 + x^{2}}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}} \]

Values of f and f' for x4x^{4} / atan(x)

xf(x)f'(x)
-2-14.45153626.292487
-1-1.27323954.2823887
0.50.134800650.84581365
11.27323954.2823887
214.45153626.292487
364.84950581.274083

Questions

What is the derivative of x4x^{4} / atan(x)?

The derivative of x4arctan⁡(x)\frac{x^{4}}{\arctan\left(x\right)} is 4x3arctan⁡(x)−x411+x2arctan⁡(x)2\frac{4 x^{3} \arctan\left(x\right) - x^{4} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

What is the second derivative of x4x^{4} / atan(x)?

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

Which rule do you use for x4x^{4} / atan(x)?

The quotient rule.

What is the slope of x4x^{4} / atan(x) at x = 1?

4.2823887.

Related derivatives

    Derivative of x^4 / atan(x): (4 * x^3 * atan(x) - x^4 * 1 / (1 + x^2)) / atan(x)^2 (with steps) | WordGPT