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y=arctg^3(ln(x^1/2)/(x+2))
Derivada de la función/ arctg/ (x+2)/ x^1/2/ y=arctg^3(ln(x^1/2)/(x+2))

Derivada de y=arctg^3(ln(x^1/2)/(x+2))

Función f() - derivada -er orden en el punto
v

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
     /   /  ___\\
    3|log\\/ x /|
atan |----------|
     \  x + 2   /
atan⁡3(log⁡(x)x+2)\operatorname{atan}^{3}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)} 
atan(log(sqrt(x))/(x + 2))^3
Gráfica
02468-8-6-4-2-10102-2
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
       /   /  ___\\ /                 /  ___\\
      2|log\\/ x /| |     1        log\\/ x /|
3*atan |----------|*|----------- - ----------|
       \  x + 2   / |2*x*(x + 2)           2 |
                    \               (x + 2)  /
----------------------------------------------
                      2/  ___\                
                   log \\/ x /                
               1 + -----------                
                            2                 
                     (x + 2)
3(−log⁡(x)(x+2)2+12x(x+2))atan⁡2(log⁡(x)x+2)1+log⁡(x)2(x+2)2\frac{3 \left(- \frac{\log{\left(\sqrt{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{2 x \left(x + 2\right)}\right) \operatorname{atan}^{2}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}}
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Segunda derivada [src] 
  /                                                                           2                         2                            \                 
  |                                                       /           /  ___\\      /           /  ___\\      /   /  ___\\           |                 
  |                                                       |  1   2*log\\/ x /|      |  1   2*log\\/ x /|      |log\\/ x /|    /  ___\|                 
  |  /          /  ___\            \     /   /  ___\\     |- - + ------------|      |- - + ------------| *atan|----------|*log\\/ x /|     /   /  ___\\
  |  |1    4*log\\/ x /       2    |     |log\\/ x /|     \  x      2 + x    /      \  x      2 + x    /      \  2 + x   /           |     |log\\/ x /|
3*|- |-- - ------------ + ---------|*atan|----------| + ------------------------- - -------------------------------------------------|*atan|----------|
  |  | 2            2     x*(2 + x)|     \  2 + x   /   /       2/  ___\\                       /       2/  ___\\                    |     \  2 + x   /
  |  \x      (2 + x)               /                    |    log \\/ x /|                       |    log \\/ x /|        2           |                 
  |                                                     |1 + -----------|*(2 + x)               |1 + -----------|*(2 + x)            |                 
  |                                                     |             2 |                       |             2 |                    |                 
  \                                                     \      (2 + x)  /                       \      (2 + x)  /                    /                 
-------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                /       2/  ___\\                                                                      
                                                                |    log \\/ x /|                                                                      
                                                              2*|1 + -----------|*(2 + x)                                                              
                                                                |             2 |                                                                      
                                                                \      (2 + x)  /
3(−(−4log⁡(x)(x+2)2+2x(x+2)+1x2)atan⁡(log⁡(x)x+2)+(2log⁡(x)x+2−1x)2(1+log⁡(x)2(x+2)2)(x+2)−(2log⁡(x)x+2−1x)2log⁡(x)atan⁡(log⁡(x)x+2)(1+log⁡(x)2(x+2)2)(x+2)2)atan⁡(log⁡(x)x+2)2(1+log⁡(x)2(x+2)2)(x+2)\frac{3 \left(- \left(- \frac{4 \log{\left(\sqrt{x} \right)}}{\left(x + 2\right)^{2}} + \frac{2}{x \left(x + 2\right)} + \frac{1}{x^{2}}\right) \operatorname{atan}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)} + \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right)^{2}}{\left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right) \left(x + 2\right)} - \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right)^{2} \log{\left(\sqrt{x} \right)} \operatorname{atan}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{\left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right) \left(x + 2\right)^{2}}\right) \operatorname{atan}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{2 \left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right) \left(x + 2\right)}
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Tercera derivada [src] 
  /     /   /  ___\\ /           /  ___\                          \                           3                           3                                      /   /  ___\\ /           /  ___\\ /          /  ___\         2/  ___\        /  ___\\     /           /  ___\\ /          /  ___\            \     /   /  ___\\                         3                                    /   /  ___\\ /           /  ___\\ /          /  ___\            \           \
  |    2|log\\/ x /| |2    12*log\\/ x /       3            6     |       /           /  ___\\        /           /  ___\\       /   /  ___\\                   2|log\\/ x /| |  1   2*log\\/ x /| |1    2*log\\/ x /   12*log \\/ x /   8*log\\/ x /|     |  1   2*log\\/ x /| |1    4*log\\/ x /       2    |     |log\\/ x /|     /           /  ___\\      /   /  ___\\                  2|log\\/ x /| |  1   2*log\\/ x /| |1    4*log\\/ x /       2    |    /  ___\|
  |atan |----------|*|-- - ------------- + ---------- + ----------|       |  1   2*log\\/ x /|        |  1   2*log\\/ x /|      2|log\\/ x /|    2/  ___\   atan |----------|*|- - + ------------|*|-- - ------------ + -------------- - ------------|   3*|- - + ------------|*|-- - ------------ + ---------|*atan|----------|     |  1   2*log\\/ x /|      |log\\/ x /|    /  ___\   atan |----------|*|- - + ------------|*|-- - ------------ + ---------|*log\\/ x /|
  |     \  2 + x   / | 3             3      2                    2|       |- - + ------------|        |- - + ------------| *atan |----------|*log \\/ x /        \  2 + x   / \  x      2 + x    / | 2         2                  2       x*(2 + x)  |     \  x      2 + x    / | 2            2     x*(2 + x)|     \  2 + x   /   3*|- - + ------------| *atan|----------|*log\\/ x /        \  2 + x   / \  x      2 + x    / | 2            2     x*(2 + x)|           |
  |                  \x       (2 + x)      x *(2 + x)   x*(2 + x) /       \  x      2 + x    /        \  x      2 + x    /       \  2 + x   /                                                      \x         x            (2 + x)                   /                          \x      (2 + x)               /                      \  x      2 + x    /      \  2 + x   /                                                     \x      (2 + x)               /           |
3*|---------------------------------------------------------------- - ----------------------------- - --------------------------------------------------- + ------------------------------------------------------------------------------------------ + ----------------------------------------------------------------------- + --------------------------------------------------- - ---------------------------------------------------------------------------------|
  |                               2                                                      2                                         2                                                         /       2/  ___\\                                                                   /       2/  ___\\                                                               2                                                   /       2/  ___\\                                    |
  |                                                                     /       2/  ___\\                         /       2/  ___\\                                                          |    log \\/ x /|        2                                                          |    log \\/ x /|                                              /       2/  ___\\                                                    |    log \\/ x /|        2                           |
  |                                                                     |    log \\/ x /|         2               |    log \\/ x /|         4                                              4*|1 + -----------|*(2 + x)                                                         2*|1 + -----------|*(2 + x)                                      |    log \\/ x /|         3                                          |1 + -----------|*(2 + x)                            |
  |                                                                   4*|1 + -----------| *(2 + x)                |1 + -----------| *(2 + x)                                                 |             2 |                                                                   |             2 |                                            2*|1 + -----------| *(2 + x)                                           |             2 |                                    |
  |                                                                     |             2 |                         |             2 |                                                          \      (2 + x)  /                                                                   \      (2 + x)  /                                              |             2 |                                                    \      (2 + x)  /                                    |
  \                                                                     \      (2 + x)  /                         \      (2 + x)  /                                                                                                                                                                                                             \      (2 + x)  /                                                                                                         /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                         /       2/  ___\\                                                                                                                                                                                                                                 
                                                                                                                                                                                                                         |    log \\/ x /|                                                                                                                                                                                                                                 
                                                                                                                                                                                                                         |1 + -----------|*(2 + x)                                                                                                                                                                                                                         
                                                                                                                                                                                                                         |             2 |                                                                                                                                                                                                                                 
                                                                                                                                                                                                                         \      (2 + x)  /
3((−12log⁡(x)(x+2)3+6x(x+2)2+3x2(x+2)+2x3)atan⁡2(log⁡(x)x+2)2+3(2log⁡(x)x+2−1x)(−4log⁡(x)(x+2)2+2x(x+2)+1x2)atan⁡(log⁡(x)x+2)2(1+log⁡(x)2(x+2)2)(x+2)−(2log⁡(x)x+2−1x)(−4log⁡(x)(x+2)2+2x(x+2)+1x2)log⁡(x)atan⁡2(log⁡(x)x+2)(1+log⁡(x)2(x+2)2)(x+2)2+(2log⁡(x)x+2−1x)(12log⁡(x)2(x+2)2−8log⁡(x)x(x+2)−2log⁡(x)x2+1x2)atan⁡2(log⁡(x)x+2)4(1+log⁡(x)2(x+2)2)(x+2)2−(2log⁡(x)x+2−1x)34(1+log⁡(x)2(x+2)2)2(x+2)2+3(2log⁡(x)x+2−1x)3log⁡(x)atan⁡(log⁡(x)x+2)2(1+log⁡(x)2(x+2)2)2(x+2)3−(2log⁡(x)x+2−1x)3log⁡(x)2atan⁡2(log⁡(x)x+2)(1+log⁡(x)2(x+2)2)2(x+2)4)(1+log⁡(x)2(x+2)2)(x+2)\frac{3 \left(\frac{\left(- \frac{12 \log{\left(\sqrt{x} \right)}}{\left(x + 2\right)^{3}} + \frac{6}{x \left(x + 2\right)^{2}} + \frac{3}{x^{2} \left(x + 2\right)} + \frac{2}{x^{3}}\right) \operatorname{atan}^{2}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{2} + \frac{3 \left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right) \left(- \frac{4 \log{\left(\sqrt{x} \right)}}{\left(x + 2\right)^{2}} + \frac{2}{x \left(x + 2\right)} + \frac{1}{x^{2}}\right) \operatorname{atan}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{2 \left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right) \left(x + 2\right)} - \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right) \left(- \frac{4 \log{\left(\sqrt{x} \right)}}{\left(x + 2\right)^{2}} + \frac{2}{x \left(x + 2\right)} + \frac{1}{x^{2}}\right) \log{\left(\sqrt{x} \right)} \operatorname{atan}^{2}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{\left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right) \left(x + 2\right)^{2}} + \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right) \left(\frac{12 \log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}} - \frac{8 \log{\left(\sqrt{x} \right)}}{x \left(x + 2\right)} - \frac{2 \log{\left(\sqrt{x} \right)}}{x^{2}} + \frac{1}{x^{2}}\right) \operatorname{atan}^{2}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{4 \left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right) \left(x + 2\right)^{2}} - \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right)^{3}}{4 \left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right)^{2} \left(x + 2\right)^{2}} + \frac{3 \left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right)^{3} \log{\left(\sqrt{x} \right)} \operatorname{atan}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{2 \left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right)^{2} \left(x + 2\right)^{3}} - \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{x + 2} - \frac{1}{x}\right)^{3} \log{\left(\sqrt{x} \right)}^{2} \operatorname{atan}^{2}{\left(\frac{\log{\left(\sqrt{x} \right)}}{x + 2} \right)}}{\left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right)^{2} \left(x + 2\right)^{4}}\right)}{\left(1 + \frac{\log{\left(\sqrt{x} \right)}^{2}}{\left(x + 2\right)^{2}}\right) \left(x + 2\right)}
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Gráfico
Derivada de y=arctg^3(ln(x^1/2)/(x+2))
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