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How to use it?
- Derivative of:
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Derivative of (x^2+1)/x
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Derivative of x^2*atanh(x)*acoth(x)
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Derivative of sqrt(x^2)
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Derivative of coth(x)/tanh(x)
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Identical expressions
- coth(x)/tanh(x)
- hyperbolic co tangent of gent of (x) divide by hyperbolic tangent of gent of (x)
- cothx/tanhx
- coth(x) divide by tanh(x)
Derivative of coth(x)/tanh(x)
The solution
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coth(x) ------- tanh(x)
$$\frac{\coth{\left(x \right)}}{\tanh{\left(x \right)}}$$
coth(x)/tanh(x)
The first derivative
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/ 2 \
1 \-1 + tanh (x)/*coth(x)
- ---------------- + -----------------------
2 2
sinh (x)*tanh(x) tanh (x)
$$\frac{\left(\tanh^{2}{\left(x \right)} - 1\right) \coth{\left(x \right)}}{\tanh^{2}{\left(x \right)}} - \frac{1}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}}$$
The second derivative
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/ / 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)
$$\frac{2 \left(\left(\frac{\tanh^{2}{\left(x \right)} - 1}{\tanh^{2}{\left(x \right)}} - 1\right) \left(\tanh^{2}{\left(x \right)} - 1\right) \coth{\left(x \right)} - \frac{\tanh^{2}{\left(x \right)} - 1}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}} + \frac{\cosh{\left(x \right)}}{\sinh^{3}{\left(x \right)}}\right)}{\tanh{\left(x \right)}}$$
The third derivative
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/ 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) /
$$2 \left(\frac{1 - \frac{3 \cosh^{2}{\left(x \right)}}{\sinh^{2}{\left(x \right)}}}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}} - \frac{3 \left(\frac{\tanh^{2}{\left(x \right)} - 1}{\tanh^{2}{\left(x \right)}} - 1\right) \left(\tanh^{2}{\left(x \right)} - 1\right)}{\sinh^{2}{\left(x \right)} \tanh{\left(x \right)}} + \frac{3 \left(\tanh^{2}{\left(x \right)} - 1\right) \cosh{\left(x \right)}}{\sinh^{3}{\left(x \right)} \tanh^{2}{\left(x \right)}} + \left(\frac{3 \left(\tanh^{2}{\left(x \right)} - 1\right)^{3}}{\tanh^{4}{\left(x \right)}} - \frac{5 \left(\tanh^{2}{\left(x \right)} - 1\right)^{2}}{\tanh^{2}{\left(x \right)}} + 2 \tanh^{2}{\left(x \right)} - 2\right) \coth{\left(x \right)}\right)$$
The graph