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

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

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

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Graph of f(x) = cos(x) / x^2 and its derivative-3-2-1123-2020406080slope -1.922f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(x) / x^2 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.9221.

How to work it out

  1. 1

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

    f(x)=cos⁡(x)x2f(x) = \frac{\cos\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=cos⁡(x)u = \cos\left(x\right) and v=x2v = x^{2}.

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=−(sin⁡(x)x2)−cos⁡(x)2x(x2)2f'(x) = \frac{-\left(\sin\left(x\right) x^{2}\right) - \cos\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)=(−(cos⁡(x)x2+sin⁡(x)2x)−(−(sin⁡(x)2x)+2cos⁡(x)))(x2)2−(−(sin⁡(x)x2)−cos⁡(x)2x)2x22x((x2)2)2f''(x) = \frac{\left(-\left(\cos\left(x\right) x^{2} + \sin\left(x\right) 2 x\right) - \left(-\left(\sin\left(x\right) 2 x\right) + 2 \cos\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(-\left(\sin\left(x\right) x^{2}\right) - \cos\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
\[ \frac{d}{dx}\left[\frac{\cos\left(x\right)}{x^{2}}\right] = \frac{-\left(\sin\left(x\right) x^{2}\right) - \cos\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f(x) = \frac{\cos\left(x\right)}{x^{2}} \]
\[ u = \cos\left(x\right),\quad v = x^{2} \]
\[ u' = -\sin\left(x\right),\quad v' = 2 x \]
\[ f'(x) = \frac{\left(-\sin\left(x\right)\right)\left(x^{2}\right) - \left(\cos\left(x\right)\right)\left(2 x\right)}{\left(x^{2}\right)^{2}} \]
\[ f'(x) = \frac{-\left(\sin\left(x\right) x^{2}\right) - \cos\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{\left(-\left(\cos\left(x\right) x^{2} + \sin\left(x\right) 2 x\right) - \left(-\left(\sin\left(x\right) 2 x\right) + 2 \cos\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(-\left(\sin\left(x\right) x^{2}\right) - \cos\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-0.104036710.12328765
-10.540302311.9220756
0.53.5103302-15.959023
10.54030231-1.9220756
2-0.10403671-0.12328765
3-0.109999170.057652777

Questions

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

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

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

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

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

The quotient rule.

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

-1.9220756.

Related derivatives

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