Производная log(sin(x),cos(x))

Вы ввели

log(sin(x), cos(x))

$$\log{\left (\sin{\left (x \right )},\cos{\left (x \right )} \right )}$$

Первая производная

cos(x)   /  d  /log(sin(x))\\|                  
------ - |-----|-----------|||           *sin(x)
sin(x)   \dxi_2\ log(xi_2) //|xi_2=cos(x)       

$$- \sin{\left (x \right )} \left. \frac{\partial}{\partial \xi_{2}}\left(\frac{\log{\left (\sin{\left (x \right )} \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=\cos{\left (x \right )} }} + \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$$

Вторая производная

                                                         /         /         2     \                   \       
                                                         |         |1 + -----------|*log(sin(x))*sin(x)|       
                                                         |  1      \    log(cos(x))/                   |       
                                                         |------ + ------------------------------------|*sin(x)
        2                                                |sin(x)                    2                  |       
     cos (x)   /  d  /log(sin(x))\\|                     \                       cos (x)               /       
-1 - ------- - |-----|-----------|||           *cos(x) + ------------------------------------------------------
        2      \dxi_2\ log(xi_2) //|xi_2=cos(x)                                  2                             
     sin (x)                                                                  log (cos(x))                     

$$\frac{\sin{\left (x \right )}}{\log^{2}{\left (\cos{\left (x \right )} \right )}} \left(\frac{\log{\left (\sin{\left (x \right )} \right )}}{\cos^{2}{\left (x \right )}} \left(1 + \frac{2}{\log{\left (\cos{\left (x \right )} \right )}}\right) \sin{\left (x \right )} + \frac{1}{\sin{\left (x \right )}}\right) - \cos{\left (x \right )} \left. \frac{\partial}{\partial \xi_{2}}\left(\frac{\log{\left (\sin{\left (x \right )} \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=\cos{\left (x \right )} }} - 1 - \frac{\cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}}$$

Третья производная

                                                                 /         2                                       /         2     \                                            2    /         2     \                    2    /         2     \            \            /         /         2     \                   \       
                                                                 |1 + -----------                                  |1 + -----------|*log(sin(x))        2                  2*sin (x)*|1 + -----------|*log(sin(x))   2*sin (x)*|1 + -----------|*log(sin(x))|            |         |1 + -----------|*log(sin(x))*sin(x)|       
                                                                 |    log(cos(x))    cos(x)           2            \    log(cos(x))/               2*sin (x)*log(sin(x))             \    log(cos(x))/                         \    log(cos(x))/            |            |  1      \    log(cos(x))/                   |       
                                                                 |--------------- - ------- + ------------------ + ----------------------------- + --------------------- + --------------------------------------- + ---------------------------------------|*sin(x)   2*|------ + ------------------------------------|*cos(x)
                                               3                 |     cos(x)          2      cos(x)*log(cos(x))               cos(x)                  3       2                              3                                   3                         |            |sin(x)                    2                  |       
/  d  /log(sin(x))\\|                     2*cos (x)   2*cos(x)   \                  sin (x)                                                         cos (x)*log (cos(x))                   cos (x)                             cos (x)*log(cos(x))          /            \                       cos (x)               /       
|-----|-----------|||           *sin(x) + --------- + -------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------
\dxi_2\ log(xi_2) //|xi_2=cos(x)              3        sin(x)                                                                                                   2                                                                                                                               2                              
                                           sin (x)                                                                                                           log (cos(x))                                                                                                                    log (cos(x))                      

$$\frac{2 \cos{\left (x \right )}}{\log^{2}{\left (\cos{\left (x \right )} \right )}} \left(\frac{\log{\left (\sin{\left (x \right )} \right )}}{\cos^{2}{\left (x \right )}} \left(1 + \frac{2}{\log{\left (\cos{\left (x \right )} \right )}}\right) \sin{\left (x \right )} + \frac{1}{\sin{\left (x \right )}}\right) + \frac{\sin{\left (x \right )}}{\log^{2}{\left (\cos{\left (x \right )} \right )}} \left(\frac{2 \sin^{2}{\left (x \right )}}{\cos^{3}{\left (x \right )}} \left(1 + \frac{2}{\log{\left (\cos{\left (x \right )} \right )}}\right) \log{\left (\sin{\left (x \right )} \right )} + \frac{\log{\left (\sin{\left (x \right )} \right )}}{\cos{\left (x \right )}} \left(1 + \frac{2}{\log{\left (\cos{\left (x \right )} \right )}}\right) + \frac{2 \left(1 + \frac{2}{\log{\left (\cos{\left (x \right )} \right )}}\right) \log{\left (\sin{\left (x \right )} \right )} \sin^{2}{\left (x \right )}}{\log{\left (\cos{\left (x \right )} \right )} \cos^{3}{\left (x \right )}} + \frac{1}{\cos{\left (x \right )}} \left(1 + \frac{2}{\log{\left (\cos{\left (x \right )} \right )}}\right) + \frac{2 \log{\left (\sin{\left (x \right )} \right )} \sin^{2}{\left (x \right )}}{\log^{2}{\left (\cos{\left (x \right )} \right )} \cos^{3}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{\sin^{2}{\left (x \right )}} + \frac{2}{\log{\left (\cos{\left (x \right )} \right )} \cos{\left (x \right )}}\right) + \sin{\left (x \right )} \left. \frac{\partial}{\partial \xi_{2}}\left(\frac{\log{\left (\sin{\left (x \right )} \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=\cos{\left (x \right )} }} + \frac{2 \cos{\left (x \right )}}{\sin{\left (x \right )}} + \frac{2 \cos^{3}{\left (x \right )}}{\sin^{3}{\left (x \right )}}$$

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