Derivative of 1/tgx*lnx

The solution

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

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

log(x)/tan(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
To find $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                                                                 /            2   \
                                                                                                   /       2   \ |     1 + tan (x)|
    /                               2                  3\                                        6*\1 + tan (x)/*|-1 + -----------|
    |                  /       2   \      /       2   \ |                        /       2   \                   |          2     |
    |         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |              2       3*\1 + tan (x)/                   \       tan (x)  /
- 2*|2 + 2*tan (x) - ---------------- + ----------------|*log(x) + --------- + --------------- + ----------------------------------
    |                       2                  4        |           3              2    2                     x*tan(x)             
    \                    tan (x)            tan (x)     /          x *tan(x)      x *tan (x)

$$- 2 \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(x \right)}} - \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right) \log{\left(x \right)} + \frac{6 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x \tan{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2} \tan^{2}{\left(x \right)}} + \frac{2}{x^{3} \tan{\left(x \right)}}$$

Simplify

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Derivative of 1/tgx*lnx

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The solution