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Identical expressions
- x^tgx*arctgx^ two
- x to the power of tgx multiply by arctgx squared
- x to the power of tgx multiply by arctgx to the power of two
- xtgx*arctgx2
- x^tgx*arctgx²
- x to the power of tgx*arctgx to the power of 2
- x^tgxarctgx^2
- xtgxarctgx2
Derivative of x^tgx*arctgx^2
The solution
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tan(x) 2 x *acot (x)
$$x^{\tan{\left(x \right)}} \operatorname{acot}^{2}{\left(x \right)}$$
d / tan(x) 2 \ --\x *acot (x)/ dx
$$\frac{d}{d x} x^{\tan{\left(x \right)}} \operatorname{acot}^{2}{\left(x \right)}$$
The first derivative
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tan(x)
tan(x) 2 /tan(x) / 2 \ \ 2*x *acot(x)
x *acot (x)*|------ + \1 + tan (x)/*log(x)| - -----------------
\ x / 2
1 + x
$$x^{\tan{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right) \operatorname{acot}^{2}{\left(x \right)} - \frac{2 x^{\tan{\left(x \right)}} \operatorname{acot}{\left(x \right)}}{x^{2} + 1}$$
The second derivative
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/ /tan(x) / 2 \ \ \
| / 2 / 2 \ \ 4*|------ + \1 + tan (x)/*log(x)|*acot(x)|
tan(x) | 2 |/tan(x) / 2 \ \ tan(x) 2*\1 + tan (x)/ / 2 \ | 2*(1 + 2*x*acot(x)) \ x / |
x *|acot (x)*||------ + \1 + tan (x)/*log(x)| - ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)| + ------------------- - -----------------------------------------|
| |\ x / 2 x | 2 2 |
| \ x / / 2\ 1 + x |
\ \1 + x / /
$$x^{\tan{\left(x \right)}} \left(\left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right) \operatorname{acot}^{2}{\left(x \right)} - \frac{4 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right) \operatorname{acot}{\left(x \right)}}{x^{2} + 1} + \frac{2 \cdot \left(2 x \operatorname{acot}{\left(x \right)} + 1\right)}{\left(x^{2} + 1\right)^{2}}\right)$$
The third derivative
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/ / 2 \ / 2 / 2 \ \ \
| | 3*x 4*x *acot(x)| |/tan(x) / 2 \ \ tan(x) 2*\1 + tan (x)/ / 2 \ | |
| 4*|-acot(x) + ------ + ------------| 6*||------ + \1 + tan (x)/*log(x)| - ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*acot(x) /tan(x) / 2 \ \|
| / 3 / 2 \ 2 / / 2 \ \ / 2 \ \ | 2 2 | |\ x / 2 x | 6*(1 + 2*x*acot(x))*|------ + \1 + tan (x)/*log(x)||
tan(x) | 2 |/tan(x) / 2 \ \ 3*\1 + tan (x)/ 2*tan(x) / 2 \ /tan(x) / 2 \ \ | tan(x) 2*\1 + tan (x)/ / 2 \ | 2 / 2 \ 6*\1 + tan (x)/*tan(x)| \ 1 + x 1 + x / \ x / \ x /|
x *|acot (x)*||------ + \1 + tan (x)/*log(x)| - --------------- + -------- + 2*\1 + tan (x)/ *log(x) + 3*|------ + \1 + tan (x)/*log(x)|*|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)| + 4*tan (x)*\1 + tan (x)/*log(x) + ----------------------| - ------------------------------------ - ------------------------------------------------------------------------------------------------------- + ---------------------------------------------------|
| |\ x / 2 3 \ x / | 2 x | x | 2 2 2 |
| \ x x \ x / / / 2\ 1 + x / 2\ |
\ \1 + x / \1 + x / /
$$x^{\tan{\left(x \right)}} \left(\left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)^{3} + 3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right) + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan^{2}{\left(x \right)} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{2 \tan{\left(x \right)}}{x^{3}}\right) \operatorname{acot}^{2}{\left(x \right)} - \frac{6 \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right) \operatorname{acot}{\left(x \right)}}{x^{2} + 1} + \frac{6 \cdot \left(2 x \operatorname{acot}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)}{\left(x^{2} + 1\right)^{2}} - \frac{4 \cdot \left(\frac{4 x^{2} \operatorname{acot}{\left(x \right)}}{x^{2} + 1} + \frac{3 x}{x^{2} + 1} - \operatorname{acot}{\left(x \right)}\right)}{\left(x^{2} + 1\right)^{2}}\right)$$
The graph