Derivative of tg(x)/ln(x)

The solution

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tan(x)
------
log(x)

$$\frac{\tan{\left(x \right)}}{\log{\left(x \right)}}$$

tan(x)/log(x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \tan{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

$\frac{\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\tan{\left(x \right)}}{x}}{\log{\left(x \right)}^{2}}$
Now simplify:

$\frac{x \log{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}}{x \log{\left(x \right)}^{2} \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{x \log{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}}{x \log{\left(x \right)}^{2} \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2               
1 + tan (x)     tan(x) 
----------- - ---------
   log(x)          2   
              x*log (x)

$$\frac{\tan^{2}{\left(x \right)} + 1}{\log{\left(x \right)}} - \frac{\tan{\left(x \right)}}{x \log{\left(x \right)}^{2}}$$

Simplify

The second derivative [src]

                                           /      2   \       
                           /       2   \   |1 + ------|*tan(x)
  /       2   \          2*\1 + tan (x)/   \    log(x)/       
2*\1 + tan (x)/*tan(x) - --------------- + -------------------
                             x*log(x)            2            
                                                x *log(x)     
--------------------------------------------------------------
                            log(x)

$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \tan{\left(x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}$$

Simplify

The third derivative [src]

                                                             /      3         3   \                                      
                                                           2*|1 + ------ + -------|*tan(x)     /       2   \ /      2   \
                                    /       2   \            |    log(x)      2   |          3*\1 + tan (x)/*|1 + ------|
  /       2   \ /         2   \   6*\1 + tan (x)/*tan(x)     \             log (x)/                          \    log(x)/
2*\1 + tan (x)/*\1 + 3*tan (x)/ - ---------------------- - ------------------------------- + ----------------------------
                                         x*log(x)                      3                               2                 
                                                                      x *log(x)                       x *log(x)          
-------------------------------------------------------------------------------------------------------------------------
                                                          log(x)

$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \tan{\left(x \right)}}{x^{3} \log{\left(x \right)}}}{\log{\left(x \right)}}$$

Simplify

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Derivative of tg(x)/ln(x)

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