Mister Exam

Derivative of ln(ch(x))/ln(cox(x))

Function f() - derivative -N order at the point
v

The graph:

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Piecewise:

The solution

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 log(cosh(x))
-------------
log(cos(x)*x)
$$\frac{\log{\left(\cosh{\left(x \right)} \right)}}{\log{\left(x \cos{\left(x \right)} \right)}}$$
log(cosh(x))/log(cos(x)*x)
The graph
The first derivative [src]
       sinh(x)          (-x*sin(x) + cos(x))*log(cosh(x))
--------------------- - ---------------------------------
cosh(x)*log(cos(x)*x)                    2               
                             x*cos(x)*log (cos(x)*x)     
$$\frac{\sinh{\left(x \right)}}{\log{\left(x \cos{\left(x \right)} \right)} \cosh{\left(x \right)}} - \frac{\left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(\cosh{\left(x \right)} \right)}}{x \log{\left(x \cos{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}$$
The second derivative [src]
               /                                                                                               2\                                              
               |                      -cos(x) + x*sin(x)   (-cos(x) + x*sin(x))*sin(x)   2*(-cos(x) + x*sin(x)) |                                              
        2      |2*sin(x) + x*cos(x) - ------------------ + --------------------------- + -----------------------|*log(cosh(x))                                 
    sinh (x)   \                              x                       cos(x)              x*cos(x)*log(x*cos(x))/                2*(-cos(x) + x*sin(x))*sinh(x)
1 - -------- + --------------------------------------------------------------------------------------------------------------- + ------------------------------
        2                                                   x*cos(x)*log(x*cos(x))                                               x*cos(x)*cosh(x)*log(x*cos(x))
    cosh (x)                                                                                                                                                   
---------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                         log(x*cos(x))                                                                         
$$\frac{- \frac{\sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} + 1 + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sinh{\left(x \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} \cosh{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} - \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right) \log{\left(\cosh{\left(x \right)} \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}}}{\log{\left(x \cos{\left(x \right)} \right)}}$$
The third derivative [src]
  /         2   \             /         2   \                          /                                                                             2                                                                               2                          3                          2                                                               \                                                                                                                            
  |     sinh (x)|             |     sinh (x)|                          |-cos(x) + x*sin(x)   2*sin(x) + x*cos(x)   (2*sin(x) + x*cos(x))*sin(x)   sin (x)*(-cos(x) + x*sin(x))   (-cos(x) + x*sin(x))*sin(x)   3*(-cos(x) + x*sin(x))     3*(-cos(x) + x*sin(x))     3*(-cos(x) + x*sin(x)) *sin(x)   3*(-cos(x) + x*sin(x))*(2*sin(x) + x*cos(x))         |                  /                                                                                               2\        
2*|-1 + --------|*sinh(x)   3*|-1 + --------|*(-cos(x) + x*sin(x))   2*|------------------ - ------------------- + ---------------------------- + ---------------------------- - --------------------------- - ----------------------- + ------------------------- + ------------------------------ + -------------------------------------------- + cos(x)|*log(cosh(x))     |                      -cos(x) + x*sin(x)   (-cos(x) + x*sin(x))*sin(x)   2*(-cos(x) + x*sin(x)) |        
  |         2   |             |         2   |                          |         2                    x                       cos(x)                           2                           x*cos(x)             2                         2    2       2                     2                                   x*cos(x)*log(x*cos(x))                    |                3*|2*sin(x) + x*cos(x) - ------------------ + --------------------------- + -----------------------|*sinh(x)
  \     cosh (x)/             \     cosh (x)/                          \        x                                                                           cos (x)                                            x *cos(x)*log(x*cos(x))   x *cos (x)*log (x*cos(x))      x*cos (x)*log(x*cos(x))                                                            /                  \                              x                       cos(x)              x*cos(x)*log(x*cos(x))/        
------------------------- - -------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------
         cosh(x)                    x*cos(x)*log(x*cos(x))                                                                                                                                                  x*cos(x)*log(x*cos(x))                                                                                                                                                                                 x*cos(x)*cosh(x)*log(x*cos(x))                                       
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                     log(x*cos(x))                                                                                                                                                                                                                                      
$$\frac{\frac{2 \left(\frac{\sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - 1\right) \sinh{\left(x \right)}}{\cosh{\left(x \right)}} - \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\frac{\sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - 1\right)}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} + \frac{3 \left(x \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} - \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right) \sinh{\left(x \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} \cosh{\left(x \right)}} + \frac{2 \left(\frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \cos{\left(x \right)} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos^{2}{\left(x \right)}} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} - \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{3}}{x^{2} \log{\left(x \cos{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)}} - \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x^{2} \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} + \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x^{2}}\right) \log{\left(\cosh{\left(x \right)} \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}}}{\log{\left(x \cos{\left(x \right)} \right)}}$$