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Identical expressions
- cth4x^ five *arccos2x
- cth4x to the power of 5 multiply by arc co sinus of e of 2x
- cth4x to the power of five multiply by arc co sinus of e of 2x
- cth4x5*arccos2x
- cth4x⁵*arccos2x
- cth4x^5arccos2x
- cth4x5arccos2x
Derivative of cth4x^5*arccos2x
The solution
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[src]
5 coth (4*x)*acos(2*x)
$$\coth^{5}{\left(4 x \right)} \operatorname{acos}{\left(2 x \right)}$$
d / 5 \ --\coth (4*x)*acos(2*x)/ dx
$$\frac{d}{d x} \coth^{5}{\left(4 x \right)} \operatorname{acos}{\left(2 x \right)}$$
The first derivative
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5 4
2*coth (4*x) 20*coth (4*x)*acos(2*x)
- ------------- - -----------------------
__________ 2
/ 2 sinh (4*x)
\/ 1 - 4*x
$$- \frac{2 \coth^{5}{\left(4 x \right)}}{\sqrt{- 4 x^{2} + 1}} - \frac{20 \coth^{4}{\left(4 x \right)} \operatorname{acos}{\left(2 x \right)}}{\sinh^{2}{\left(4 x \right)}}$$
The second derivative
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/ / 2 \ \
| 2 20*|--------- + cosh(4*x)*coth(4*x)|*acos(2*x)|
3 | x*coth (4*x) 10*coth(4*x) \sinh(4*x) / |
8*coth (4*x)*|- ------------- + ------------------------ + ----------------------------------------------|
| 3/2 __________ 3 |
| / 2\ / 2 2 sinh (4*x) |
\ \1 - 4*x / \/ 1 - 4*x *sinh (4*x) /
$$8 \left(- \frac{x \coth^{2}{\left(4 x \right)}}{\left(- 4 x^{2} + 1\right)^{\frac{3}{2}}} + \frac{20 \left(\cosh{\left(4 x \right)} \coth{\left(4 x \right)} + \frac{2}{\sinh{\left(4 x \right)}}\right) \operatorname{acos}{\left(2 x \right)}}{\sinh^{3}{\left(4 x \right)}} + \frac{10 \coth{\left(4 x \right)}}{\sqrt{- 4 x^{2} + 1} \sinh^{2}{\left(4 x \right)}}\right) \coth^{3}{\left(4 x \right)}$$
The third derivative
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/ / 2 \ / 2 2 \ \
| 3 | 12*x | | 2 6 3*cosh (4*x)*coth (4*x) 12*cosh(4*x)*coth(4*x)| |
|coth (4*x)*|-1 + ---------| 80*|- coth (4*x) + ---------- + ----------------------- + ----------------------|*acos(2*x) / 2 \ |
| | 2| | 4 2 3 | 120*|--------- + cosh(4*x)*coth(4*x)|*coth(4*x) 2 |
2 | \ -1 + 4*x / \ sinh (4*x) sinh (4*x) sinh (4*x) / \sinh(4*x) / 60*x*coth (4*x) |
8*coth (4*x)*|--------------------------- - ------------------------------------------------------------------------------------------- - ----------------------------------------------- + ------------------------|
| 3/2 2 __________ 3/2 |
| / 2\ sinh (4*x) / 2 3 / 2\ 2 |
\ \1 - 4*x / \/ 1 - 4*x *sinh (4*x) \1 - 4*x / *sinh (4*x)/
$$8 \cdot \left(\frac{\left(\frac{12 x^{2}}{4 x^{2} - 1} - 1\right) \coth^{3}{\left(4 x \right)}}{\left(- 4 x^{2} + 1\right)^{\frac{3}{2}}} - \frac{80 \left(- \coth^{2}{\left(4 x \right)} + \frac{3 \cosh^{2}{\left(4 x \right)} \coth^{2}{\left(4 x \right)}}{\sinh^{2}{\left(4 x \right)}} + \frac{12 \cosh{\left(4 x \right)} \coth{\left(4 x \right)}}{\sinh^{3}{\left(4 x \right)}} + \frac{6}{\sinh^{4}{\left(4 x \right)}}\right) \operatorname{acos}{\left(2 x \right)}}{\sinh^{2}{\left(4 x \right)}} + \frac{60 x \coth^{2}{\left(4 x \right)}}{\left(- 4 x^{2} + 1\right)^{\frac{3}{2}} \sinh^{2}{\left(4 x \right)}} - \frac{120 \left(\cosh{\left(4 x \right)} \coth{\left(4 x \right)} + \frac{2}{\sinh{\left(4 x \right)}}\right) \coth{\left(4 x \right)}}{\sqrt{- 4 x^{2} + 1} \sinh^{3}{\left(4 x \right)}}\right) \coth^{2}{\left(4 x \right)}$$
The graph