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

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

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

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Graph of f(x) = tan(x) / (1 / x) and its derivative-3-2-1123-40-30-20-1010203040slope 4.983f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(x) / (1 / x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 4.9829.

How to work it out

  1. 1

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

    f(x)=tan⁡(x)1xf(x) = \frac{\tan\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=tan⁡(x)u = \tan\left(x\right) and v=1xv = \frac{1}{x}.

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=1xcos⁡(x)2−tan⁡(x)−1x21x2f'(x) = \frac{\frac{\frac{1}{x}}{\cos\left(x\right)^{2}} - \tan\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)=(−1x2cos⁡(x)2−2cos⁡(x)−sin⁡(x)x(cos⁡(x)2)2−(−1x2cos⁡(x)2+tan⁡(x)−((−2)x)(x2)2))1x2−(1xcos⁡(x)2−tan⁡(x)−1x2)21x−1x2(1x2)2f''(x) = \frac{\left(\frac{\frac{-1}{x^{2}} \cos\left(x\right)^{2} - \frac{2 \cos\left(x\right) -\sin\left(x\right)}{x}}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{\frac{-1}{x^{2}}}{\cos\left(x\right)^{2}} + \tan\left(x\right) \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\frac{\frac{1}{x}}{\cos\left(x\right)^{2}} - \tan\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
\[ \frac{d}{dx}\left[\frac{\tan\left(x\right)}{\frac{1}{x}}\right] = \frac{\frac{\frac{1}{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f(x) = \frac{\tan\left(x\right)}{\frac{1}{x}} \]
\[ u = \tan\left(x\right),\quad v = \frac{1}{x} \]
\[ u' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(\frac{1}{x}\right) - \left(\tan\left(x\right)\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}} \]
\[ f'(x) = \frac{\frac{\frac{1}{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f''(x) = \frac{\left(\frac{\frac{-1}{x^{2}} \cos\left(x\right)^{2} - \frac{2 \cos\left(x\right) -\sin\left(x\right)}{x}}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{\frac{-1}{x^{2}}}{\cos\left(x\right)^{2}} + \tan\left(x\right) \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(\frac{\frac{1}{x}}{\cos\left(x\right)^{2}} - \tan\left(x\right) \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-4.3700797-9.3637585
-11.5574077-4.9829265
0.50.273151241.1955257
11.55740774.9829265
2-4.37007979.3637585
3-0.427639632.918412

Questions

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

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

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

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

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

The quotient rule.

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

4.9829265.

Related derivatives

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