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- Derivative of:
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Derivative of y=x/lnx
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Identical expressions
- y=lnsin^ two *arctgx^ three
- y equally ln sinus of squared multiply by arctgx cubed
- y equally ln sinus of to the power of two multiply by arctgx to the power of three
- y=lnsin2*arctgx3
- y=lnsin²*arctgx³
- y=lnsin to the power of 2*arctgx to the power of 3
- y=lnsin^2arctgx^3
- y=lnsin2arctgx3
Derivative of y=lnsin^2*arctgx^3
The solution
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2 3 log (sin(acot(x)))*x
$$x^{3} \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}^{2}$$
log(sin(acot(x)))^2*x^3
The first derivative
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3
2 2 2*x *log(sin(acot(x)))
3*x *log (sin(acot(x))) - -----------------------------------
________
/ 1 / 2\
/ 1 + -- *\1 + x /*sin(acot(x))
/ 2
\/ x
$$- \frac{2 x^{3} \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\sqrt{1 + \frac{1}{x^{2}}} \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + 3 x^{2} \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}^{2}$$
The second derivative
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/ 2 / 1 log(sin(acot(x))) log(sin(acot(x))) 2*x*log(sin(acot(x))) \ \
| x *|------------------------------ - ----------------- - ------------------------------ + ----------------------| |
| |/ 1 \ / 2\ 3/2 / 1 \ / 2\ ________ | |
| ||1 + --|*\1 + x /*sin(acot(x)) 3 / 1 \ |1 + --|*\1 + x /*sin(acot(x)) / 1 / 2\| |
| || 2| x *|1 + --| | 2| / 1 + -- *\1 + x /| |
| |\ x / | 2| \ x / / 2 | |
| 2 \ \ x / \/ x / 6*x*log(sin(acot(x))) |
2*x*|3*log (sin(acot(x))) + ----------------------------------------------------------------------------------------------------------------- - -----------------------------------|
| / 2\ ________ |
| \1 + x /*sin(acot(x)) / 1 / 2\ |
| / 1 + -- *\1 + x /*sin(acot(x))|
| / 2 |
\ \/ x /
$$2 x \left(\frac{x^{2} \left(\frac{2 x \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\sqrt{1 + \frac{1}{x^{2}}} \left(x^{2} + 1\right)} - \frac{\log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\left(1 + \frac{1}{x^{2}}\right) \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{1}{\left(1 + \frac{1}{x^{2}}\right) \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} - \frac{\log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}}}\right)}{\left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} - \frac{6 x \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\sqrt{1 + \frac{1}{x^{2}}} \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + 3 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}^{2}\right)$$
The third derivative
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/ / 2 \ \ | 3 | 3*log(sin(acot(x))) 2*log(sin(acot(x))) 3*log(sin(acot(x))) 3 8*x *log(sin(acot(x))) 6*x 2*log(sin(acot(x))) 3 4*log(sin(acot(x))) 3*log(sin(acot(x))) 6*x*log(sin(acot(x))) | 2 / 1 log(sin(acot(x))) log(sin(acot(x))) 2*x*log(sin(acot(x))) \ | | x *|- ------------------- + ---------------------- + ------------------- + ----------------------------------- - ----------------------- - ------------------------------- - ----------------------------------- + ---------------------------------- + ----------------------- - ---------------------------------- + -------------------------------| 9*x *|------------------------------ - ----------------- - ------------------------------ + ----------------------| | | | 5/2 ________ 3/2 3/2 2 ________ 2 2 3/2 2 2 3/2 2 2 | |/ 1 \ / 2\ 3/2 / 1 \ / 2\ ________ | | | | 6 / 1 \ / 1 / 2\ 4 / 1 \ / 1 \ / 2\ 2 / 1 / 2\ / 1 \ / 2\ / 1 \ / 2\ 2 3 / 1 \ / 2\ 2 / 1 \ / 2\ 3 / 1 \ / 2\ / 1 \ / 2\ | ||1 + --|*\1 + x /*sin(acot(x)) 3 / 1 \ |1 + --|*\1 + x /*sin(acot(x)) / 1 / 2\| | | | x *|1 + --| / 1 + -- *\1 + x / x *|1 + --| |1 + --| *\1 + x / *sin (acot(x)) / 1 + -- *\1 + x / |1 + --|*\1 + x / *sin(acot(x)) |1 + --| *\1 + x / *sin (acot(x)) x *|1 + --| *\1 + x /*sin(acot(x)) x *|1 + --| *\1 + x / x *|1 + --| *\1 + x /*sin(acot(x)) |1 + --|*\1 + x / *sin(acot(x))| || 2| x *|1 + --| | 2| / 1 + -- *\1 + x /| | | | | 2| / 2 | 2| | 2| / 2 | 2| | 2| | 2| | 2| | 2| | 2| | |\ x / | 2| \ x / / 2 | | | 2 \ \ x / \/ x \ x / \ x / \/ x \ x / \ x / \ x / \ x / \ x / \ x / / \ \ x / \/ x / 18*x*log(sin(acot(x))) | 2*|3*log (sin(acot(x))) + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------- - -----------------------------------| | / 2\ / 2\ ________ | | \1 + x /*sin(acot(x)) \1 + x /*sin(acot(x)) / 1 / 2\ | | / 1 + -- *\1 + x /*sin(acot(x))| | / 2 | \ \/ x /
$$2 \left(\frac{x^{3} \left(- \frac{8 x^{2} \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\sqrt{1 + \frac{1}{x^{2}}} \left(x^{2} + 1\right)^{2}} + \frac{6 x \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\left(1 + \frac{1}{x^{2}}\right) \left(x^{2} + 1\right)^{2} \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} - \frac{6 x}{\left(1 + \frac{1}{x^{2}}\right) \left(x^{2} + 1\right)^{2} \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{2 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\sqrt{1 + \frac{1}{x^{2}}} \left(x^{2} + 1\right)} - \frac{2 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}} \left(x^{2} + 1\right)^{2} \sin^{2}{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{3}{\left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}} \left(x^{2} + 1\right)^{2} \sin^{2}{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{4 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}} \left(x^{2} + 1\right)} - \frac{3 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)^{2} \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{3}{x^{3} \left(1 + \frac{1}{x^{2}}\right)^{2} \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{3 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{x^{4} \left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}}} - \frac{3 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{x^{6} \left(1 + \frac{1}{x^{2}}\right)^{\frac{5}{2}}}\right)}{\left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{9 x^{2} \left(\frac{2 x \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\sqrt{1 + \frac{1}{x^{2}}} \left(x^{2} + 1\right)} - \frac{\log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\left(1 + \frac{1}{x^{2}}\right) \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + \frac{1}{\left(1 + \frac{1}{x^{2}}\right) \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} - \frac{\log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}}}\right)}{\left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} - \frac{18 x \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}}{\sqrt{1 + \frac{1}{x^{2}}} \left(x^{2} + 1\right) \sin{\left(\operatorname{acot}{\left(x \right)} \right)}} + 3 \log{\left(\sin{\left(\operatorname{acot}{\left(x \right)} \right)} \right)}^{2}\right)$$
The graph