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Derivative of sin⁡(x)x3\frac{\sin\left(x\right)}{x^{3}}

The derivative of sin⁡(x)x3\frac{\sin\left(x\right)}{x^{3}} is cos⁡(x)x3−sin⁡(x)3x2(x3)2\frac{\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}.

f(x) =
ddx[sin⁡(x)x3]=cos⁡(x)x3−sin⁡(x)3x2(x3)2\frac{d}{dx}\left[\frac{\sin\left(x\right)}{x^{3}}\right] = \frac{\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}

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Graph of f(x) = sin(x) / x^3 and its derivative-3-2-1123-2020406080slope -1.984f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) / x^3 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.9841.

How to work it out

  1. 1

    We differentiate f(x)=sin⁡(x)x3f(x) = \frac{\sin\left(x\right)}{x^{3}} with respect to x. The rule that applies is the quotient rule.

    f(x)=sin⁡(x)x3f(x) = \frac{\sin\left(x\right)}{x^{3}}
  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=x3v = x^{3}.

    u=sin⁡(x),v=x3u = \sin\left(x\right),\quad v = x^{3}
  3. 3

    Differentiate each piece:

    u′=cos⁡(x),v′=3x2u' = \cos\left(x\right),\quad v' = 3 x^{2}
  4. 4

    Put them in the rule:

    f′(x)=(cos⁡(x))(x3)−(sin⁡(x))(3x2)(x3)2f'(x) = \frac{\left(\cos\left(x\right)\right)\left(x^{3}\right) - \left(\sin\left(x\right)\right)\left(3 x^{2}\right)}{\left(x^{3}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=cos⁡(x)x3−sin⁡(x)3x2(x3)2f'(x) = \frac{\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}
  6. 6

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

    f′′(x)=(−(sin⁡(x)x3)+cos⁡(x)3x2−(cos⁡(x)3x2+sin⁡(x)6x))(x3)2−(cos⁡(x)x3−sin⁡(x)3x2)2x33x2((x3)2)2f''(x) = \frac{\left(-\left(\sin\left(x\right) x^{3}\right) + \cos\left(x\right) 3 x^{2} - \left(\cos\left(x\right) 3 x^{2} + \sin\left(x\right) 6 x\right)\right) \left(x^{3}\right)^{2} - \left(\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}\right) 2 x^{3} 3 x^{2}}{\left(\left(x^{3}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\sin\left(x\right)}{x^{3}}\right] = \frac{\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}} \]
\[ f(x) = \frac{\sin\left(x\right)}{x^{3}} \]
\[ u = \sin\left(x\right),\quad v = x^{3} \]
\[ u' = \cos\left(x\right),\quad v' = 3 x^{2} \]
\[ f'(x) = \frac{\left(\cos\left(x\right)\right)\left(x^{3}\right) - \left(\sin\left(x\right)\right)\left(3 x^{2}\right)}{\left(x^{3}\right)^{2}} \]
\[ f'(x) = \frac{\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}} \]
\[ f''(x) = \frac{\left(-\left(\sin\left(x\right) x^{3}\right) + \cos\left(x\right) 3 x^{2} - \left(\cos\left(x\right) 3 x^{2} + \sin\left(x\right) 6 x\right)\right) \left(x^{3}\right)^{2} - \left(\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}\right) 2 x^{3} 3 x^{2}}{\left(\left(x^{3}\right)^{2}\right)^{2}} \]

Values of f and f' for sin(x) / x3x^{3}

xf(x)f'(x)
-20.113662180.22251162
-10.841470981.9841106
0.53.8354043-15.991765
10.84147098-1.9841106
20.11366218-0.22251162
30.005226667-0.041893056

Questions

What is the derivative of sin(x) / x3x^{3}?

The derivative of sin⁡(x)x3\frac{\sin\left(x\right)}{x^{3}} is cos⁡(x)x3−sin⁡(x)3x2(x3)2\frac{\cos\left(x\right) x^{3} - \sin\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}.

What is the second derivative of sin(x) / x3x^{3}?

((-(sin(x) * x3x^{3}) + cos(x) * 3 * x2x^{2} - (cos(x) * 3 * x2x^{2} + sin(x) * 6x)) * (x3x^{3})^2 - (cos(x) * x3x^{3} - sin(x) * 3 * x2x^{2}) * 2 * x3x^{3} * 3 * x2x^{2}) / ((x3x^{3})^2)^2.

Which rule do you use for sin(x) / x3x^{3}?

The quotient rule.

What is the slope of sin(x) / x3x^{3} at x = 1?

-1.9841106.

Related derivatives

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