Производная log(x,cos(x))

Вы ввели

log(x, cos(x))

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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