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coth(x)/tanh(x)
The derivative of the function/ coth(x)/tanh(x)

Derivative of coth(x)/tanh(x)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
coth(x)
-------
tanh(x)
coth⁡(x)tanh⁡(x)\frac{\coth{\left(x \right)}}{\tanh{\left(x \right)}} 
coth(x)/tanh(x)
The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                     /         2   \        
         1           \-1 + tanh (x)/*coth(x)
- ---------------- + -----------------------
      2                          2          
  sinh (x)*tanh(x)           tanh (x)
(tanh⁡2(x)−1)coth⁡(x)tanh⁡2(x)−1sinh⁡2(x)tanh⁡(x)\frac{\left(\tanh^{2}{\left(x \right)} - 1\right) \coth{\left(x \right)}}{\tanh^{2}{\left(x \right)}} - \frac{1}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}}
Simplify
The second derivative [src] 
  /                           /              2   \                     2     \
  |cosh(x)    /         2   \ |     -1 + tanh (x)|            -1 + tanh (x)  |
2*|-------- + \-1 + tanh (x)/*|-1 + -------------|*coth(x) - ----------------|
  |    3                      |            2     |               2           |
  \sinh (x)                   \        tanh (x)  /           sinh (x)*tanh(x)/
------------------------------------------------------------------------------
                                   tanh(x)
2((tanh⁡2(x)−1tanh⁡2(x)−1)(tanh⁡2(x)−1)coth⁡(x)−tanh⁡2(x)−1sinh⁡2(x)tanh⁡(x)+cosh⁡(x)sinh⁡3(x))tanh⁡(x)\frac{2 \left(\left(\frac{\tanh^{2}{\left(x \right)} - 1}{\tanh^{2}{\left(x \right)}} - 1\right) \left(\tanh^{2}{\left(x \right)} - 1\right) \coth{\left(x \right)} - \frac{\tanh^{2}{\left(x \right)} - 1}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}} + \frac{\cosh{\left(x \right)}}{\sinh^{3}{\left(x \right)}}\right)}{\tanh{\left(x \right)}}
Simplify
The third derivative [src] 
  /                                                                                 2                         /              2   \                            \
  |                                                                           3*cosh (x)      /         2   \ |     -1 + tanh (x)|                            |
  |/                                   2                    3\            1 - ----------    3*\-1 + tanh (x)/*|-1 + -------------|                            |
  ||                    /         2   \      /         2   \ |                     2                          |            2     |     /         2   \        |
  ||           2      5*\-1 + tanh (x)/    3*\-1 + tanh (x)/ |                 sinh (x)                       \        tanh (x)  /   3*\-1 + tanh (x)/*cosh(x)|
2*||-2 + 2*tanh (x) - ------------------ + ------------------|*coth(x) + ---------------- - -------------------------------------- + -------------------------|
  ||                           2                    4        |               2                             2                                 3        2       |
  \\                       tanh (x)             tanh (x)     /           sinh (x)*tanh(x)              sinh (x)*tanh(x)                  sinh (x)*tanh (x)    /
2(1−3cosh⁡2(x)sinh⁡2(x)sinh⁡2(x)tanh⁡(x)−3(tanh⁡2(x)−1tanh⁡2(x)−1)(tanh⁡2(x)−1)sinh⁡2(x)tanh⁡(x)+3(tanh⁡2(x)−1)cosh⁡(x)sinh⁡3(x)tanh⁡2(x)+(3(tanh⁡2(x)−1)3tanh⁡4(x)−5(tanh⁡2(x)−1)2tanh⁡2(x)+2tanh⁡2(x)−2)coth⁡(x))2 \left(\frac{1 - \frac{3 \cosh^{2}{\left(x \right)}}{\sinh^{2}{\left(x \right)}}}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}} - \frac{3 \left(\frac{\tanh^{2}{\left(x \right)} - 1}{\tanh^{2}{\left(x \right)}} - 1\right) \left(\tanh^{2}{\left(x \right)} - 1\right)}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}} + \frac{3 \left(\tanh^{2}{\left(x \right)} - 1\right) \cosh{\left(x \right)}}{\sinh^{3}{\left(x \right)} \tanh^{2}{\left(x \right)}} + \left(\frac{3 \left(\tanh^{2}{\left(x \right)} - 1\right)^{3}}{\tanh^{4}{\left(x \right)}} - \frac{5 \left(\tanh^{2}{\left(x \right)} - 1\right)^{2}}{\tanh^{2}{\left(x \right)}} + 2 \tanh^{2}{\left(x \right)} - 2\right) \coth{\left(x \right)}\right)
Simplify
The graph
Derivative of coth(x)/tanh(x)
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