Производная (1/2)*tan(x)*acot(x)

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↑ Функция f () ? - производная -го порядка

Решение

Вы ввели
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tan(x)        
------*acot(x)
  2           
$$\frac{1}{2} \tan{\left (x \right )} \operatorname{acot}{\left (x \right )}$$
График
Первая производная
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/       2   \                     
|1   tan (x)|             tan(x)  
|- + -------|*acot(x) - ----------
\2      2   /             /     2\
                        2*\1 + x /
$$\left(\frac{1}{2} \tan^{2}{\left (x \right )} + \frac{1}{2}\right) \operatorname{acot}{\left (x \right )} - \frac{\tan{\left (x \right )}}{2 x^{2} + 2}$$
Вторая производная
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         2                                              
  1 + tan (x)    x*tan(x)   /       2   \               
- ----------- + --------- + \1 + tan (x)/*acot(x)*tan(x)
          2             2                               
     1 + x      /     2\                                
                \1 + x /                                
$$\frac{x \tan{\left (x \right )}}{\left(x^{2} + 1\right)^{2}} + \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} \operatorname{acot}{\left (x \right )} - \frac{\tan^{2}{\left (x \right )} + 1}{x^{2} + 1}$$
Третья производная
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                         2              2            /       2   \                                                /       2   \
  tan(x)    /       2   \            4*x *tan(x)   3*\1 + tan (x)/*tan(x)        2    /       2   \           3*x*\1 + tan (x)/
--------- + \1 + tan (x)/ *acot(x) - ----------- - ---------------------- + 2*tan (x)*\1 + tan (x)/*acot(x) + -----------------
        2                                     3                 2                                                         2    
/     2\                              /     2\             1 + x                                                  /     2\     
\1 + x /                              \1 + x /                                                                    \1 + x /     
$$- \frac{4 x^{2} \tan{\left (x \right )}}{\left(x^{2} + 1\right)^{3}} + \frac{3 x}{\left(x^{2} + 1\right)^{2}} \left(\tan^{2}{\left (x \right )} + 1\right) + \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \operatorname{acot}{\left (x \right )} + 2 \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (x \right )} \operatorname{acot}{\left (x \right )} - \frac{3 \tan{\left (x \right )}}{x^{2} + 1} \left(\tan^{2}{\left (x \right )} + 1\right) + \frac{\tan{\left (x \right )}}{\left(x^{2} + 1\right)^{2}}$$