Mister Exam

Derivative of cosx/lnx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
cos(x)
------
log(x)
$$\frac{\cos{\left(x \right)}}{\log{\left(x \right)}}$$
cos(x)/log(x)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of cosine is negative sine:

    To find :

    1. The derivative of is .

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

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