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

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

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

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Graph of f(x) = sin(x) / x^4 and its derivative-3-2-1123-1000-800-600-400-200slope -2.826f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) / x^4 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -2.8256.

How to work it out

  1. 1

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

    f(x)=sin⁡(x)x4f(x) = \frac{\sin\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=sin⁡(x)u = \sin\left(x\right) and v=x4v = x^{4}.

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

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

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

xf(x)f'(x)
-2-0.056831089-0.13967136
-1-0.84147098-2.8255816
0.57.6708086-47.325148
10.84147098-2.8255816
20.056831089-0.13967136
30.0017422223-0.014545093

Questions

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

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

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

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

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

The quotient rule.

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

-2.8255816.

Related derivatives

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