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How to use it?
- Derivative of:
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Derivative of 5-7x
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Derivative of 2*sqrt(x^3)
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Derivative of ln(x^2-2x+2)
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Derivative of (2x-1)/(2x+1)
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Identical expressions
- th(x)/ch^ two (x)
- th(x) divide by ch squared (x)
- th(x) divide by ch to the power of two (x)
- th(x)/ch2(x)
- thx/ch2x
- th(x)/ch²(x)
- th(x)/ch to the power of 2(x)
- thx/ch^2x
- th(x) divide by ch^2(x)
Derivative of th(x)/ch^2(x)
The solution
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tanh(x)
--------
2
cosh (x)
$$\frac{\tanh{\left(x \right)}}{\cosh^{2}{\left(x \right)}}$$
d /tanh(x) \ --|--------| dx| 2 | \cosh (x)/
$$\frac{d}{d x} \frac{\tanh{\left(x \right)}}{\cosh^{2}{\left(x \right)}}$$
The first derivative
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2
1 - tanh (x) 2*sinh(x)*tanh(x)
------------ - -----------------
2 3
cosh (x) cosh (x)
$$\frac{1 - \tanh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - \frac{2 \sinh{\left(x \right)} \tanh{\left(x \right)}}{\cosh^{3}{\left(x \right)}}$$
The second derivative
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/ / 2 \ / 2 \ \
|/ 2 \ | 3*sinh (x)| 2*\-1 + tanh (x)/*sinh(x)|
2*|\-1 + tanh (x)/*tanh(x) + |-1 + ----------|*tanh(x) + -------------------------|
| | 2 | cosh(x) |
\ \ cosh (x) / /
-----------------------------------------------------------------------------------
2
cosh (x)
$$\frac{2 \left(\left(\frac{3 \sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - 1\right) \tanh{\left(x \right)} + \frac{2 \left(\tanh^{2}{\left(x \right)} - 1\right) \sinh{\left(x \right)}}{\cosh{\left(x \right)}} + \left(\tanh^{2}{\left(x \right)} - 1\right) \tanh{\left(x \right)}\right)}{\cosh^{2}{\left(x \right)}}$$
The third derivative
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/ / 2 \ \
| | 3*sinh (x)| |
| 4*|-2 + ----------|*sinh(x)*tanh(x) |
| / 2 \ | 2 | / 2 \ |
|/ 2 \ / 2 \ / 2 \ | 3*sinh (x)| \ cosh (x) / 6*\-1 + tanh (x)/*sinh(x)*tanh(x)|
-2*|\-1 + tanh (x)/*\-1 + 3*tanh (x)/ + 3*\-1 + tanh (x)/*|-1 + ----------| + ----------------------------------- + ---------------------------------|
| | 2 | cosh(x) cosh(x) |
\ \ cosh (x) / /
------------------------------------------------------------------------------------------------------------------------------------------------------
2
cosh (x)
$$- \frac{2 \cdot \left(\frac{4 \cdot \left(\frac{3 \sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - 2\right) \sinh{\left(x \right)} \tanh{\left(x \right)}}{\cosh{\left(x \right)}} + 3 \cdot \left(\frac{3 \sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - 1\right) \left(\tanh^{2}{\left(x \right)} - 1\right) + \left(\tanh^{2}{\left(x \right)} - 1\right) \left(3 \tanh^{2}{\left(x \right)} - 1\right) + \frac{6 \left(\tanh^{2}{\left(x \right)} - 1\right) \sinh{\left(x \right)} \tanh{\left(x \right)}}{\cosh{\left(x \right)}}\right)}{\cosh^{2}{\left(x \right)}}$$
The graph