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Derivative of sin⁡(x)x2\frac{\sin\left(x\right)}{x^{2}}

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

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

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Graph of f(x) = sin(x) / x^2 and its derivative-3-2-1123-15-10-55slope -1.143f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) / x^2 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.1426.

How to work it out

  1. 1

    We differentiate f(x)=sin⁡(x)x2f(x) = \frac{\sin\left(x\right)}{x^{2}} with respect to x. The rule that applies is the quotient rule.

    f(x)=sin⁡(x)x2f(x) = \frac{\sin\left(x\right)}{x^{2}}
  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=x2v = x^{2}.

    u=sin⁡(x),v=x2u = \sin\left(x\right),\quad v = x^{2}
  3. 3

    Differentiate each piece:

    u′=cos⁡(x),v′=2xu' = \cos\left(x\right),\quad v' = 2 x
  4. 4

    Put them in the rule:

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

    Simplify:

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

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

    f′′(x)=(−(sin⁡(x)x2)+cos⁡(x)2x−(cos⁡(x)2x+2sin⁡(x)))(x2)2−(cos⁡(x)x2−sin⁡(x)2x)2x22x((x2)2)2f''(x) = \frac{\left(-\left(\sin\left(x\right) x^{2}\right) + \cos\left(x\right) 2 x - \left(\cos\left(x\right) 2 x + 2 \sin\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\cos\left(x\right) x^{2} - \sin\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\sin\left(x\right)}{x^{2}}\right] = \frac{\cos\left(x\right) x^{2} - \sin\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f(x) = \frac{\sin\left(x\right)}{x^{2}} \]
\[ u = \sin\left(x\right),\quad v = x^{2} \]
\[ u' = \cos\left(x\right),\quad v' = 2 x \]
\[ f'(x) = \frac{\left(\cos\left(x\right)\right)\left(x^{2}\right) - \left(\sin\left(x\right)\right)\left(2 x\right)}{\left(x^{2}\right)^{2}} \]
\[ f'(x) = \frac{\cos\left(x\right) x^{2} - \sin\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{\left(-\left(\sin\left(x\right) x^{2}\right) + \cos\left(x\right) 2 x - \left(\cos\left(x\right) 2 x + 2 \sin\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\cos\left(x\right) x^{2} - \sin\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}} \]

Values of f and f' for sin(x) / x2x^{2}

xf(x)f'(x)
-2-0.22732436-0.33136107
-1-0.84147098-1.1426397
0.51.9177022-4.1604784
10.84147098-1.1426397
20.22732436-0.33136107
30.015680001-0.1204525

Questions

What is the derivative of sin(x) / x2x^{2}?

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

What is the second derivative of sin(x) / x2x^{2}?

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

Which rule do you use for sin(x) / x2x^{2}?

The quotient rule.

What is the slope of sin(x) / x2x^{2} at x = 1?

-1.1426397.

Related derivatives

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