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Derivative of sin⁡(x)1x\frac{\sin\left(x\right)}{\frac{1}{x}}

The derivative of sin⁡(x)1x\frac{\sin\left(x\right)}{\frac{1}{x}} is cos⁡(x)1x−sin⁡(x)−1x21x2\frac{\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

f(x) =
ddx[sin⁡(x)1x]=cos⁡(x)1x−sin⁡(x)−1x21x2\frac{d}{dx}\left[\frac{\sin\left(x\right)}{\frac{1}{x}}\right] = \frac{\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}

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Graph of f(x) = sin(x) / (1 / x) and its derivative-3-2-1123-4-3-2-1123slope 1.382f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) / (1 / x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.3818.

How to work it out

  1. 1

    We differentiate f(x)=sin⁡(x)1xf(x) = \frac{\sin\left(x\right)}{\frac{1}{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=sin⁡(x)1xf(x) = \frac{\sin\left(x\right)}{\frac{1}{x}}
  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=1xv = \frac{1}{x}.

    u=sin⁡(x),v=1xu = \sin\left(x\right),\quad v = \frac{1}{x}
  3. 3

    Differentiate each piece:

    u′=cos⁡(x),v′=−1x2u' = \cos\left(x\right),\quad v' = \frac{-1}{x^{2}}
  4. 4

    Put them in the rule:

    f′(x)=(cos⁡(x))(1x)−(sin⁡(x))(−1x2)(1x)2f'(x) = \frac{\left(\cos\left(x\right)\right)\left(\frac{1}{x}\right) - \left(\sin\left(x\right)\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=cos⁡(x)1x−sin⁡(x)−1x21x2f'(x) = \frac{\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}
  6. 6

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

    f′′(x)=(−(sin⁡(x)1x)+cos⁡(x)−1x2−(cos⁡(x)−1x2+sin⁡(x)−((−2)x)(x2)2))1x2−(cos⁡(x)1x−sin⁡(x)−1x2)21x−1x2(1x2)2f''(x) = \frac{\left(-\left(\sin\left(x\right) \frac{1}{x}\right) + \cos\left(x\right) \frac{-1}{x^{2}} - \left(\cos\left(x\right) \frac{-1}{x^{2}} + \sin\left(x\right) \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\sin\left(x\right)}{\frac{1}{x}}\right] = \frac{\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f(x) = \frac{\sin\left(x\right)}{\frac{1}{x}} \]
\[ u = \sin\left(x\right),\quad v = \frac{1}{x} \]
\[ u' = \cos\left(x\right),\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \frac{\left(\cos\left(x\right)\right)\left(\frac{1}{x}\right) - \left(\sin\left(x\right)\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}} \]
\[ f'(x) = \frac{\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f''(x) = \frac{\left(-\left(\sin\left(x\right) \frac{1}{x}\right) + \cos\left(x\right) \frac{-1}{x^{2}} - \left(\cos\left(x\right) \frac{-1}{x^{2}} + \sin\left(x\right) \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}} \]

Values of f and f' for sin(x) / (1 / x)

xf(x)f'(x)
-21.8185949-0.077003754
-10.84147098-1.3817733
0.50.239712770.91821682
10.841470981.3817733
21.81859490.077003754
30.42336002-2.8288575

Questions

What is the derivative of sin(x) / (1 / x)?

The derivative of sin⁡(x)1x\frac{\sin\left(x\right)}{\frac{1}{x}} is cos⁡(x)1x−sin⁡(x)−1x21x2\frac{\cos\left(x\right) \frac{1}{x} - \sin\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

What is the second derivative of sin(x) / (1 / x)?

((-(sin(x) * 1 / x) + cos(x) * -1 / x2x^{2} - (cos(x) * -1 / x2x^{2} + sin(x) * -(-2x) / (x2x^{2})^2)) * (1 / x)^2 - (cos(x) * 1 / x - sin(x) * -1 / x2x^{2}) * 2 * 1 / x * -1 / x2x^{2}) / ((1 / x)^2)^2.

Which rule do you use for sin(x) / (1 / x)?

The quotient rule.

What is the slope of sin(x) / (1 / x) at x = 1?

1.3817733.

Related derivatives

    Derivative of sin(x) / (1 / x): (cos(x) * 1 / x - sin(x) * -1 / x^2) / (1 / x)^2 (with steps) | WordGPT