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y=(ln(arctg(3/x)))^(-1)
The derivative of the function/ y=(ln(arctg(3/x)))^(-1)

Derivative of y=(ln(arctg(3/x)))^(-1)

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

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The solution

You have entered [src] 
     1      
------------
   /    /3\\
log|atan|-||
   \    \x//
1log⁡(atan⁡(3x))\frac{1}{\log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)}} 
1/log(atan(3/x))
The graph
02468-8-6-4-2-1010-100000100000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                3                
---------------------------------
 2 /    9 \     /3\    2/    /3\\
x *|1 + --|*atan|-|*log |atan|-||
   |     2|     \x/     \    \x//
   \    x /
3x2(1+9x2)log⁡(atan⁡(3x))2atan⁡(3x)\frac{3}{x^{2} \left(1 + \frac{9}{x^{2}}\right) \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)}^{2} \operatorname{atan}{\left(\frac{3}{x} \right)}}
Simplify
The second derivative [src] 
  /          18               3                           6               \
3*|-2 + ----------- + ------------------ + -------------------------------|
  |      2 /    9 \     /    9 \     /3\     /    9 \     /3\    /    /3\\|
  |     x *|1 + --|   x*|1 + --|*atan|-|   x*|1 + --|*atan|-|*log|atan|-|||
  |        |     2|     |     2|     \x/     |     2|     \x/    \    \x//|
  \        \    x /     \    x /             \    x /                     /
---------------------------------------------------------------------------
                      3 /    9 \     /3\    2/    /3\\                     
                     x *|1 + --|*atan|-|*log |atan|-||                     
                        |     2|     \x/     \    \x//                     
                        \    x /
3(−2+3x(1+9x2)atan⁡(3x)+6x(1+9x2)log⁡(atan⁡(3x))atan⁡(3x)+18x2(1+9x2))x3(1+9x2)log⁡(atan⁡(3x))2atan⁡(3x)\frac{3 \left(-2 + \frac{3}{x \left(1 + \frac{9}{x^{2}}\right) \operatorname{atan}{\left(\frac{3}{x} \right)}} + \frac{6}{x \left(1 + \frac{9}{x^{2}}\right) \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)} \operatorname{atan}{\left(\frac{3}{x} \right)}} + \frac{18}{x^{2} \left(1 + \frac{9}{x^{2}}\right)}\right)}{x^{3} \left(1 + \frac{9}{x^{2}}\right) \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)}^{2} \operatorname{atan}{\left(\frac{3}{x} \right)}}
Simplify
The third derivative [src] 
   /         21           108                3                      3                      27                           6                                  9                                     9                                    54               \
18*|1 - ----------- + ------------ - ------------------ + --------------------- + -------------------- - ------------------------------- + ---------------------------------- + ----------------------------------- + ---------------------------------|
   |     2 /    9 \              2     /    9 \     /3\              2                       2             /    9 \     /3\    /    /3\\              2                                    2                                     2                     |
   |    x *|1 + --|    4 /    9 \    x*|1 + --|*atan|-|    2 /    9 \      2/3\    3 /    9 \      /3\   x*|1 + --|*atan|-|*log|atan|-||    2 /    9 \      2/3\    /    /3\\    2 /    9 \      2/3\    2/    /3\\    3 /    9 \      /3\    /    /3\\|
   |       |     2|   x *|1 + --|      |     2|     \x/   x *|1 + --| *atan |-|   x *|1 + --| *atan|-|     |     2|     \x/    \    \x//   x *|1 + --| *atan |-|*log|atan|-||   x *|1 + --| *atan |-|*log |atan|-||   x *|1 + --| *atan|-|*log|atan|-|||
   |       \    x /      |     2|      \    x /              |     2|       \x/      |     2|      \x/     \    x /                           |     2|       \x/    \    \x//      |     2|       \x/     \    \x//      |     2|      \x/    \    \x//|
   \                     \    x /                            \    x /                \    x /                                                 \    x /                             \    x /                              \    x /                      /
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                                                                                                            4 /    9 \     /3\    2/    /3\\                                                                                                            
                                                                                                           x *|1 + --|*atan|-|*log |atan|-||                                                                                                            
                                                                                                              |     2|     \x/     \    \x//                                                                                                            
                                                                                                              \    x /
18(1−3x(1+9x2)atan⁡(3x)−6x(1+9x2)log⁡(atan⁡(3x))atan⁡(3x)−21x2(1+9x2)+3x2(1+9x2)2atan⁡2(3x)+9x2(1+9x2)2log⁡(atan⁡(3x))atan⁡2(3x)+9x2(1+9x2)2log⁡(atan⁡(3x))2atan⁡2(3x)+27x3(1+9x2)2atan⁡(3x)+54x3(1+9x2)2log⁡(atan⁡(3x))atan⁡(3x)+108x4(1+9x2)2)x4(1+9x2)log⁡(atan⁡(3x))2atan⁡(3x)\frac{18 \left(1 - \frac{3}{x \left(1 + \frac{9}{x^{2}}\right) \operatorname{atan}{\left(\frac{3}{x} \right)}} - \frac{6}{x \left(1 + \frac{9}{x^{2}}\right) \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)} \operatorname{atan}{\left(\frac{3}{x} \right)}} - \frac{21}{x^{2} \left(1 + \frac{9}{x^{2}}\right)} + \frac{3}{x^{2} \left(1 + \frac{9}{x^{2}}\right)^{2} \operatorname{atan}^{2}{\left(\frac{3}{x} \right)}} + \frac{9}{x^{2} \left(1 + \frac{9}{x^{2}}\right)^{2} \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)} \operatorname{atan}^{2}{\left(\frac{3}{x} \right)}} + \frac{9}{x^{2} \left(1 + \frac{9}{x^{2}}\right)^{2} \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)}^{2} \operatorname{atan}^{2}{\left(\frac{3}{x} \right)}} + \frac{27}{x^{3} \left(1 + \frac{9}{x^{2}}\right)^{2} \operatorname{atan}{\left(\frac{3}{x} \right)}} + \frac{54}{x^{3} \left(1 + \frac{9}{x^{2}}\right)^{2} \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)} \operatorname{atan}{\left(\frac{3}{x} \right)}} + \frac{108}{x^{4} \left(1 + \frac{9}{x^{2}}\right)^{2}}\right)}{x^{4} \left(1 + \frac{9}{x^{2}}\right) \log{\left(\operatorname{atan}{\left(\frac{3}{x} \right)} \right)}^{2} \operatorname{atan}{\left(\frac{3}{x} \right)}}
Simplify
The graph
Derivative of y=(ln(arctg(3/x)))^(-1)
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