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Derivative of cos⁡(x)x4\frac{\cos\left(x\right)}{x^{4}}

The derivative of cos⁡(x)x4\frac{\cos\left(x\right)}{x^{4}} is −(sin⁡(x)x4)−cos⁡(x)4x3(x4)2\frac{-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}.

f(x) =
ddx[cos⁡(x)x4]=−(sin⁡(x)x4)−cos⁡(x)4x3(x4)2\frac{d}{dx}\left[\frac{\cos\left(x\right)}{x^{4}}\right] = \frac{-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}

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Graph of f(x) = cos(x) / x^4 and its derivative-3-2-112350010001500200025003000slope -3.003f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(x) / x^4 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -3.0027.

How to work it out

  1. 1

    We differentiate f(x)=cos⁡(x)x4f(x) = \frac{\cos\left(x\right)}{x^{4}} with respect to x. The rule that applies is the quotient rule.

    f(x)=cos⁡(x)x4f(x) = \frac{\cos\left(x\right)}{x^{4}}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=cos⁡(x)u = \cos\left(x\right) and v=x4v = x^{4}.

    u=cos⁡(x),v=x4u = \cos\left(x\right),\quad v = x^{4}
  3. 3

    Differentiate each piece:

    u′=−sin⁡(x),v′=4x3u' = -\sin\left(x\right),\quad v' = 4 x^{3}
  4. 4

    Put them in the rule:

    f′(x)=(−sin⁡(x))(x4)−(cos⁡(x))(4x3)(x4)2f'(x) = \frac{\left(-\sin\left(x\right)\right)\left(x^{4}\right) - \left(\cos\left(x\right)\right)\left(4 x^{3}\right)}{\left(x^{4}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=−(sin⁡(x)x4)−cos⁡(x)4x3(x4)2f'(x) = \frac{-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}
  6. 6

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

    f′′(x)=(−(cos⁡(x)x4+sin⁡(x)4x3)−(−(sin⁡(x)4x3)+cos⁡(x)12x2))(x4)2−(−(sin⁡(x)x4)−cos⁡(x)4x3)2x44x3((x4)2)2f''(x) = \frac{\left(-\left(\cos\left(x\right) x^{4} + \sin\left(x\right) 4 x^{3}\right) - \left(-\left(\sin\left(x\right) 4 x^{3}\right) + \cos\left(x\right) 12 x^{2}\right)\right) \left(x^{4}\right)^{2} - \left(-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}\right) 2 x^{4} 4 x^{3}}{\left(\left(x^{4}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\cos\left(x\right)}{x^{4}}\right] = \frac{-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}} \]
\[ f(x) = \frac{\cos\left(x\right)}{x^{4}} \]
\[ u = \cos\left(x\right),\quad v = x^{4} \]
\[ u' = -\sin\left(x\right),\quad v' = 4 x^{3} \]
\[ f'(x) = \frac{\left(-\sin\left(x\right)\right)\left(x^{4}\right) - \left(\cos\left(x\right)\right)\left(4 x^{3}\right)}{\left(x^{4}\right)^{2}} \]
\[ f'(x) = \frac{-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}} \]
\[ f''(x) = \frac{\left(-\left(\cos\left(x\right) x^{4} + \sin\left(x\right) 4 x^{3}\right) - \left(-\left(\sin\left(x\right) 4 x^{3}\right) + \cos\left(x\right) 12 x^{2}\right)\right) \left(x^{4}\right)^{2} - \left(-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}\right) 2 x^{4} 4 x^{3}}{\left(\left(x^{4}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-0.0260091770.0048127346
-10.540302313.0026802
0.514.041321-120.00138
10.54030231-3.0026802
2-0.026009177-0.0048127346
3-0.012222130.01455395

Questions

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

The derivative of cos⁡(x)x4\frac{\cos\left(x\right)}{x^{4}} is −(sin⁡(x)x4)−cos⁡(x)4x3(x4)2\frac{-\left(\sin\left(x\right) x^{4}\right) - \cos\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}.

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

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

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

The quotient rule.

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

-3.0026802.

Related derivatives

    Derivative of cos(x) / x^4: (-(sin(x) * x^4) - cos(x) * 4 * x^3) / (x^4)^2 (with steps) | WordGPT