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

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

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src] 
     /   /  ___\\
    3|   |\/ x ||
atan |log|-----||
     \   \x + 2//
atan⁡3(log⁡(xx+2))\operatorname{atan}^{3}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)} 
atan(log(sqrt(x)/(x + 2)))^3
Gráfica
02468-8-6-4-2-10105-5
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Primera derivada [src] 
       /   /  ___\\         /                     ___  \
      2|   |\/ x ||         |       1           \/ x   |
3*atan |log|-----||*(x + 2)*|--------------- - --------|
       \   \x + 2//         |    ___                  2|
                            \2*\/ x *(x + 2)   (x + 2) /
--------------------------------------------------------
                      /        /  ___\\                 
                  ___ |       2|\/ x ||                 
                \/ x *|1 + log |-----||                 
                      \        \x + 2//
3(x+2)(−x(x+2)2+12x(x+2))atan⁡2(log⁡(xx+2))x(log⁡(xx+2)2+1)\frac{3 \left(x + 2\right) \left(- \frac{\sqrt{x}}{\left(x + 2\right)^{2}} + \frac{1}{2 \sqrt{x} \left(x + 2\right)}\right) \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{\sqrt{x} \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)}
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Segunda derivada [src] 
  /                                                                                                                 2                                                                 2                            \                 
  |/              ___\     /   /  ___\\   /           ___                 \     /   /  ___\\     /              ___\      /              ___\     /   /  ___\\     /              ___\      /   /  ___\\    /  ___\|                 
  ||    1     2*\/ x |     |   |\/ x ||   | 1     8*\/ x           4      |     |   |\/ x ||     |    1     2*\/ x |      |    1     2*\/ x |     |   |\/ x ||     |    1     2*\/ x |      |   |\/ x ||    |\/ x ||                 
  ||- ----- + -------|*atan|log|-----||   |---- - -------- + -------------|*atan|log|-----||   2*|- ----- + -------|    2*|- ----- + -------|*atan|log|-----||   2*|- ----- + -------| *atan|log|-----||*log|-----||                 
  ||    ___    2 + x |     \   \2 + x//   | 3/2          2     ___        |     \   \2 + x//     |    ___    2 + x |      |    ___    2 + x |     \   \2 + x//     |    ___    2 + x |      \   \2 + x//    \2 + x/|     /   /  ___\\
  |\  \/ x           /                    \x      (2 + x)    \/ x *(2 + x)/                      \  \/ x           /      \  \/ x           /                      \  \/ x           /                             |     |   |\/ x ||
3*|------------------------------------ - -------------------------------------------------- + ---------------------- - -------------------------------------- - --------------------------------------------------|*atan|log|-----||
  |                 3/2                                           ___                             /        /  ___\\                   ___                                         /        /  ___\\                |     \   \2 + x//
  |                x                                            \/ x                              |       2|\/ x ||                 \/ x *(2 + x)                                 |       2|\/ x ||                |                 
  |                                                                                             x*|1 + log |-----||                                                             x*|1 + log |-----||                |                 
  \                                                                                               \        \2 + x//                                                               \        \2 + x//                /                 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                           /        /  ___\\                                                                                                         
                                                                                                           |       2|\/ x ||                                                                                                         
                                                                                                         4*|1 + log |-----||                                                                                                         
                                                                                                           \        \2 + x//
3(−2(2xx+2−1x)2log⁡(xx+2)atan⁡(log⁡(xx+2))x(log⁡(xx+2)2+1)+2(2xx+2−1x)2x(log⁡(xx+2)2+1)−(−8x(x+2)2+4x(x+2)+1x32)atan⁡(log⁡(xx+2))x−2(2xx+2−1x)atan⁡(log⁡(xx+2))x(x+2)+(2xx+2−1x)atan⁡(log⁡(xx+2))x32)atan⁡(log⁡(xx+2))4(log⁡(xx+2)2+1)\frac{3 \left(- \frac{2 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{2} \log{\left(\frac{\sqrt{x}}{x + 2} \right)} \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{x \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} + \frac{2 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{2}}{x \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} - \frac{\left(- \frac{8 \sqrt{x}}{\left(x + 2\right)^{2}} + \frac{4}{\sqrt{x} \left(x + 2\right)} + \frac{1}{x^{\frac{3}{2}}}\right) \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{\sqrt{x}} - \frac{2 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right) \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{\sqrt{x} \left(x + 2\right)} + \frac{\left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right) \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{x^{\frac{3}{2}}}\right) \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{4 \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)}
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Tercera derivada [src] 
  /                                                                  3                                                                                                                                                                                                2                                                                                             3                                        3                                                      2                                         3                                                                                                                               2                                                     2                                                                                                                    \
  |         /   /  ___\\ /              ___\      /              ___\           /   /  ___\\ /           ___                 \          /   /  ___\\ /            ___                                \        /   /  ___\\ /              ___\     /              ___\      /   /  ___\\        /   /  ___\\ /           ___                 \   /              ___\       /   /  ___\\   /              ___\       /   /  ___\\     /  ___\     /              ___\      /   /  ___\\     /              ___\      /   /  ___\\    /  ___\     /              ___\ /           ___                 \     /   /  ___\\     /              ___\       /   /  ___\\    /  ___\     /              ___\       /   /  ___\\    /  ___\          /   /  ___\\ /              ___\ /           ___                 \    /  ___\|
  |        2|   |\/ x || |    1     2*\/ x |      |    1     2*\/ x |          2|   |\/ x || | 1     8*\/ x           4      |         2|   |\/ x || | 1     16*\/ x         2               8       |       2|   |\/ x || |    1     2*\/ x |     |    1     2*\/ x |      |   |\/ x ||       2|   |\/ x || | 1     8*\/ x           4      |   |    1     2*\/ x |      2|   |\/ x ||   |    1     2*\/ x |      2|   |\/ x ||    2|\/ x |     |    1     2*\/ x |      |   |\/ x ||     |    1     2*\/ x |      |   |\/ x ||    |\/ x |     |    1     2*\/ x | | 1     8*\/ x           4      |     |   |\/ x ||     |    1     2*\/ x |      2|   |\/ x ||    |\/ x |     |    1     2*\/ x |      2|   |\/ x ||    |\/ x |         2|   |\/ x || |    1     2*\/ x | | 1     8*\/ x           4      |    |\/ x ||
  |  3*atan |log|-----||*|- ----- + -------|      |- ----- + -------|      atan |log|-----||*|---- - -------- + -------------|   3*atan |log|-----||*|---- - -------- + ------------ + --------------|   atan |log|-----||*|- ----- + -------|   3*|- ----- + -------| *atan|log|-----||   atan |log|-----||*|---- - -------- + -------------|   |- ----- + -------| *atan |log|-----||   |- ----- + -------| *atan |log|-----||*log |-----|   3*|- ----- + -------| *atan|log|-----||   3*|- ----- + -------| *atan|log|-----||*log|-----|   3*|- ----- + -------|*|---- - -------- + -------------|*atan|log|-----||   3*|- ----- + -------| *atan |log|-----||*log|-----|   3*|- ----- + -------| *atan |log|-----||*log|-----|   3*atan |log|-----||*|- ----- + -------|*|---- - -------- + -------------|*log|-----||
  |         \   \2 + x// |    ___    2 + x |      |    ___    2 + x |           \   \2 + x// | 3/2          2     ___        |          \   \2 + x// | 5/2          3    3/2             ___        2|        \   \2 + x// |    ___    2 + x |     |    ___    2 + x |      \   \2 + x//        \   \2 + x// | 3/2          2     ___        |   |    ___    2 + x |       \   \2 + x//   |    ___    2 + x |       \   \2 + x//     \2 + x/     |    ___    2 + x |      \   \2 + x//     |    ___    2 + x |      \   \2 + x//    \2 + x/     |    ___    2 + x | | 3/2          2     ___        |     \   \2 + x//     |    ___    2 + x |       \   \2 + x//    \2 + x/     |    ___    2 + x |       \   \2 + x//    \2 + x/          \   \2 + x// |    ___    2 + x | | 3/2          2     ___        |    \2 + x/|
  |                      \  \/ x           /      \  \/ x           /                        \x      (2 + x)    \/ x *(2 + x)/                       \x      (2 + x)    x   *(2 + x)   \/ x *(2 + x) /                     \  \/ x           /     \  \/ x           /                                       \x      (2 + x)    \/ x *(2 + x)/   \  \/ x           /                      \  \/ x           /                                    \  \/ x           /                       \  \/ x           /                                  \  \/ x           / \x      (2 + x)    \/ x *(2 + x)/                      \  \/ x           /                                   \  \/ x           /                                                     \  \/ x           / \x      (2 + x)    \/ x *(2 + x)/           |
3*|- --------------------------------------- - ------------------------- + --------------------------------------------------- + --------------------------------------------------------------------- + ------------------------------------- - --------------------------------------- - --------------------------------------------------- + -------------------------------------- - -------------------------------------------------- + --------------------------------------- + -------------------------------------------------- + ------------------------------------------------------------------------ + --------------------------------------------------- - --------------------------------------------------- - ------------------------------------------------------------------------------------|
  |                      5/2                                           2                             3/2                                                            ___                                                 3/2                                    /        /  ___\\                                 ___                                           /        /  ___\\                                             2                          /        /  ___\\                                                    2                                             /        /  ___\\                                                 /        /  ___\\                                /        /  ___\\                                                         /        /  ___\\                                |
  |                   8*x                             /        /  ___\\                           4*x                                                           8*\/ x                                               2*x   *(2 + x)                          2 |       2|\/ x ||                             2*\/ x *(2 + x)                               3/2 |       2|\/ x ||                            /        /  ___\\                           |       2|\/ x ||                                   /        /  ___\\                                              |       2|\/ x ||                                               2 |       2|\/ x ||                                |       2|\/ x ||                                                         |       2|\/ x ||                                |
  |                                               3/2 |       2|\/ x ||                                                                                                                                                                                   4*x *|1 + log |-----||                                                                        4*x   *|1 + log |-----||                        3/2 |       2|\/ x ||                       2*x*|1 + log |-----||*(2 + x)                       3/2 |       2|\/ x ||                                          4*x*|1 + log |-----||                                            4*x *|1 + log |-----||                            2*x*|1 + log |-----||*(2 + x)                                             4*x*|1 + log |-----||                                |
  |                                            4*x   *|1 + log |-----||                                                                                                                                                                                        \        \2 + x//                                                                               \        \2 + x//                       x   *|1 + log |-----||                           \        \2 + x//                            2*x   *|1 + log |-----||                                              \        \2 + x//                                                 \        \2 + x//                                \        \2 + x//                                                         \        \2 + x//                                |
  \                                                   \        \2 + x//                                                                                                                                                                                                                                                                                                                                     \        \2 + x//                                                                               \        \2 + x//                                                                                                                                                                                                                                                                                            /
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                                                                                                                                                                                                                                                                                                                                                                                                             /  ___\                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                                            2|\/ x |                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                                     1 + log |-----|                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                                             \2 + x/
3(−3(2xx+2−1x)(−8x(x+2)2+4x(x+2)+1x32)log⁡(xx+2)atan⁡2(log⁡(xx+2))4x(log⁡(xx+2)2+1)+3(2xx+2−1x)(−8x(x+2)2+4x(x+2)+1x32)atan⁡(log⁡(xx+2))4x(log⁡(xx+2)2+1)−3(2xx+2−1x)2log⁡(xx+2)atan⁡2(log⁡(xx+2))2x(x+2)(log⁡(xx+2)2+1)+3(2xx+2−1x)2atan⁡(log⁡(xx+2))2x(x+2)(log⁡(xx+2)2+1)+3(2xx+2−1x)2log⁡(xx+2)atan⁡2(log⁡(xx+2))4x2(log⁡(xx+2)2+1)−3(2xx+2−1x)2atan⁡(log⁡(xx+2))4x2(log⁡(xx+2)2+1)+3(−16x(x+2)3+8x(x+2)2+2x32(x+2)+1x52)atan⁡2(log⁡(xx+2))8x−(−8x(x+2)2+4x(x+2)+1x32)atan⁡2(log⁡(xx+2))2x(x+2)+(2xx+2−1x)3atan⁡2(log⁡(xx+2))4x32(log⁡(xx+2)2+1)−(2xx+2−1x)3log⁡(xx+2)2atan⁡2(log⁡(xx+2))x32(log⁡(xx+2)2+1)2+3(2xx+2−1x)3log⁡(xx+2)atan⁡(log⁡(xx+2))2x32(log⁡(xx+2)2+1)2−(2xx+2−1x)34x32(log⁡(xx+2)2+1)2+(−8x(x+2)2+4x(x+2)+1x32)atan⁡2(log⁡(xx+2))4x32+(2xx+2−1x)atan⁡2(log⁡(xx+2))2x32(x+2)−3(2xx+2−1x)atan⁡2(log⁡(xx+2))8x52)log⁡(xx+2)2+1\frac{3 \left(- \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right) \left(- \frac{8 \sqrt{x}}{\left(x + 2\right)^{2}} + \frac{4}{\sqrt{x} \left(x + 2\right)} + \frac{1}{x^{\frac{3}{2}}}\right) \log{\left(\frac{\sqrt{x}}{x + 2} \right)} \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{4 x \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} + \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right) \left(- \frac{8 \sqrt{x}}{\left(x + 2\right)^{2}} + \frac{4}{\sqrt{x} \left(x + 2\right)} + \frac{1}{x^{\frac{3}{2}}}\right) \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{4 x \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} - \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{2} \log{\left(\frac{\sqrt{x}}{x + 2} \right)} \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{2 x \left(x + 2\right) \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} + \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{2} \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{2 x \left(x + 2\right) \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} + \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{2} \log{\left(\frac{\sqrt{x}}{x + 2} \right)} \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{4 x^{2} \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} - \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{2} \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{4 x^{2} \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} + \frac{3 \left(- \frac{16 \sqrt{x}}{\left(x + 2\right)^{3}} + \frac{8}{\sqrt{x} \left(x + 2\right)^{2}} + \frac{2}{x^{\frac{3}{2}} \left(x + 2\right)} + \frac{1}{x^{\frac{5}{2}}}\right) \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{8 \sqrt{x}} - \frac{\left(- \frac{8 \sqrt{x}}{\left(x + 2\right)^{2}} + \frac{4}{\sqrt{x} \left(x + 2\right)} + \frac{1}{x^{\frac{3}{2}}}\right) \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{2 \sqrt{x} \left(x + 2\right)} + \frac{\left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{3} \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{4 x^{\frac{3}{2}} \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)} - \frac{\left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{3} \log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{x^{\frac{3}{2}} \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)^{2}} + \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{3} \log{\left(\frac{\sqrt{x}}{x + 2} \right)} \operatorname{atan}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{2 x^{\frac{3}{2}} \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)^{2}} - \frac{\left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right)^{3}}{4 x^{\frac{3}{2}} \left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1\right)^{2}} + \frac{\left(- \frac{8 \sqrt{x}}{\left(x + 2\right)^{2}} + \frac{4}{\sqrt{x} \left(x + 2\right)} + \frac{1}{x^{\frac{3}{2}}}\right) \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{4 x^{\frac{3}{2}}} + \frac{\left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right) \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{2 x^{\frac{3}{2}} \left(x + 2\right)} - \frac{3 \left(\frac{2 \sqrt{x}}{x + 2} - \frac{1}{\sqrt{x}}\right) \operatorname{atan}^{2}{\left(\log{\left(\frac{\sqrt{x}}{x + 2} \right)} \right)}}{8 x^{\frac{5}{2}}}\right)}{\log{\left(\frac{\sqrt{x}}{x + 2} \right)}^{2} + 1}
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Gráfico
Derivada de y=arctg^3(ln(sqrt(x)/(x+2)))
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