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Derivative of extan⁡(x)\frac{e^{x}}{\tan\left(x\right)}

The derivative of extan⁡(x)\frac{e^{x}}{\tan\left(x\right)} is extan⁡(x)−ex1cos⁡(x)2tan⁡(x)2\frac{e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}}.

f(x) =
ddx[extan⁡(x)]=extan⁡(x)−ex1cos⁡(x)2tan⁡(x)2\frac{d}{dx}\left[\frac{e^{x}}{\tan\left(x\right)}\right] = \frac{e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}}

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Graph of f(x) = exp(x) / tan(x) and its derivative-3-2-1123-120-100-80-60-40-2020slope -2.094f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x) / tan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -2.0936.

How to work it out

  1. 1

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

    f(x)=extan⁡(x)f(x) = \frac{e^{x}}{\tan\left(x\right)}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=exu = e^{x} and v=tan⁡(x)v = \tan\left(x\right).

    u=ex,v=tan⁡(x)u = e^{x},\quad v = \tan\left(x\right)
  3. 3

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=extan⁡(x)−ex1cos⁡(x)2tan⁡(x)2f'(x) = \frac{e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}}
  6. 6

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

    f′′(x)=(extan⁡(x)+ex1cos⁡(x)2−(ex1cos⁡(x)2+ex−(2cos⁡(x)−sin⁡(x))(cos⁡(x)2)2))tan⁡(x)2−(extan⁡(x)−ex1cos⁡(x)2)2tan⁡(x)1cos⁡(x)2(tan⁡(x)2)2f''(x) = \frac{\left(e^{x} \tan\left(x\right) + e^{x} \frac{1}{\cos\left(x\right)^{2}} - \left(e^{x} \frac{1}{\cos\left(x\right)^{2}} + e^{x} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}}\right)\right) \tan\left(x\right)^{2} - \left(e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}\right) 2 \tan\left(x\right) \frac{1}{\cos\left(x\right)^{2}}}{\left(\tan\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{e^{x}}{\tan\left(x\right)}\right] = \frac{e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}} \]
\[ f(x) = \frac{e^{x}}{\tan\left(x\right)} \]
\[ u = e^{x},\quad v = \tan\left(x\right) \]
\[ u' = e^{x},\quad v' = \frac{1}{\cos\left(x\right)^{2}} \]
\[ f'(x) = \frac{\left(e^{x}\right)\left(\tan\left(x\right)\right) - \left(e^{x}\right)\left(\frac{1}{\cos\left(x\right)^{2}}\right)}{\left(\tan\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(e^{x} \tan\left(x\right) + e^{x} \frac{1}{\cos\left(x\right)^{2}} - \left(e^{x} \frac{1}{\cos\left(x\right)^{2}} + e^{x} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}}\right)\right) \tan\left(x\right)^{2} - \left(e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}\right) 2 \tan\left(x\right) \frac{1}{\cos\left(x\right)^{2}}}{\left(\tan\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-20.061937215-0.1017441
-1-0.23621267-0.75576253
0.53.017964-4.1551034
11.7453887-2.0935943
2-3.3816573-12.318354
3-140.90511-1149.4756

Questions

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

The derivative of extan⁡(x)\frac{e^{x}}{\tan\left(x\right)} is extan⁡(x)−ex1cos⁡(x)2tan⁡(x)2\frac{e^{x} \tan\left(x\right) - e^{x} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}}.

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

((exp(x) * tan(x) + exp(x) * 1 / cos(x)^2 - (exp(x) * 1 / cos(x)^2 + exp(x) * -(2cos(x) * -sin(x)) / (cos(x)^2)^2)) * tan(x)^2 - (exp(x) * tan(x) - exp(x) * 1 / cos(x)^2) * 2tan(x) * 1 / cos(x)^2) / (tan(x)^2)^2.

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

The quotient rule.

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

-2.0935943.

Related derivatives

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