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y=(1+x^2)^arctg(2x)

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

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

Gráfico:

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Solución

Ha introducido [src]
        atan(2*x)
/     2\         
\1 + x /         
$$\left(x^{2} + 1\right)^{\operatorname{atan}{\left(2 x \right)}}$$
(1 + x^2)^atan(2*x)
Solución detallada
  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada


Respuesta:

Gráfica
Primera derivada [src]
        atan(2*x) /     /     2\                \
/     2\          |2*log\1 + x /   2*x*atan(2*x)|
\1 + x /         *|------------- + -------------|
                  |          2              2   |
                  \   1 + 4*x          1 + x    /
$$\left(x^{2} + 1\right)^{\operatorname{atan}{\left(2 x \right)}} \left(\frac{2 x \operatorname{atan}{\left(2 x \right)}}{x^{2} + 1} + \frac{2 \log{\left(x^{2} + 1 \right)}}{4 x^{2} + 1}\right)$$
Segunda derivada [src]
                    /                             2                                                                     \
          atan(2*x) |  /   /     2\              \                       /     2\      2                                |
  /     2\          |  |log\1 + x /   x*atan(2*x)|    atan(2*x)   8*x*log\1 + x /   2*x *atan(2*x)           4*x        |
2*\1 + x /         *|2*|----------- + -----------|  + --------- - --------------- - -------------- + -------------------|
                    |  |         2            2  |           2                2               2      /     2\ /       2\|
                    |  \  1 + 4*x        1 + x   /      1 + x       /       2\        /     2\       \1 + x /*\1 + 4*x /|
                    \                                               \1 + 4*x /        \1 + x /                          /
$$2 \left(x^{2} + 1\right)^{\operatorname{atan}{\left(2 x \right)}} \left(- \frac{2 x^{2} \operatorname{atan}{\left(2 x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{8 x \log{\left(x^{2} + 1 \right)}}{\left(4 x^{2} + 1\right)^{2}} + \frac{4 x}{\left(x^{2} + 1\right) \left(4 x^{2} + 1\right)} + 2 \left(\frac{x \operatorname{atan}{\left(2 x \right)}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{4 x^{2} + 1}\right)^{2} + \frac{\operatorname{atan}{\left(2 x \right)}}{x^{2} + 1}\right)$$
3-я производная [src]
                    /                             3                                                                                                                                                                                                                                                \
          atan(2*x) |  /   /     2\              \         /     2\     /   /     2\              \ /                                       2                    /     2\\                                    2                      2                              3                 2    /     2\|
  /     2\          |  |log\1 + x /   x*atan(2*x)|    4*log\1 + x /     |log\1 + x /   x*atan(2*x)| |  atan(2*x)           4*x           2*x *atan(2*x)   8*x*log\1 + x /|            3                   24*x                    6*x            3*x*atan(2*x)   4*x *atan(2*x)   64*x *log\1 + x /|
4*\1 + x /         *|2*|----------- + -----------|  - ------------- - 3*|----------- + -----------|*|- --------- - ------------------- + -------------- + ---------------| + ------------------- - -------------------- - -------------------- - ------------- + -------------- + -----------------|
                    |  |         2            2  |               2      |         2            2  | |         2    /     2\ /       2\             2                  2  |   /     2\ /       2\                      2           2                        2               3                   3   |
                    |  \  1 + 4*x        1 + x   /     /       2\       \  1 + 4*x        1 + x   / |    1 + x     \1 + x /*\1 + 4*x /     /     2\         /       2\   |   \1 + x /*\1 + 4*x /   /     2\ /       2\    /     2\  /       2\     /     2\        /     2\          /       2\    |
                    \                                  \1 + 4*x /                                   \                                      \1 + x /         \1 + 4*x /   /                         \1 + x /*\1 + 4*x /    \1 + x / *\1 + 4*x /     \1 + x /        \1 + x /          \1 + 4*x /    /
$$4 \left(x^{2} + 1\right)^{\operatorname{atan}{\left(2 x \right)}} \left(\frac{4 x^{3} \operatorname{atan}{\left(2 x \right)}}{\left(x^{2} + 1\right)^{3}} + \frac{64 x^{2} \log{\left(x^{2} + 1 \right)}}{\left(4 x^{2} + 1\right)^{3}} - \frac{24 x^{2}}{\left(x^{2} + 1\right) \left(4 x^{2} + 1\right)^{2}} - \frac{6 x^{2}}{\left(x^{2} + 1\right)^{2} \left(4 x^{2} + 1\right)} - \frac{3 x \operatorname{atan}{\left(2 x \right)}}{\left(x^{2} + 1\right)^{2}} + 2 \left(\frac{x \operatorname{atan}{\left(2 x \right)}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{4 x^{2} + 1}\right)^{3} - 3 \left(\frac{x \operatorname{atan}{\left(2 x \right)}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{4 x^{2} + 1}\right) \left(\frac{2 x^{2} \operatorname{atan}{\left(2 x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{8 x \log{\left(x^{2} + 1 \right)}}{\left(4 x^{2} + 1\right)^{2}} - \frac{4 x}{\left(x^{2} + 1\right) \left(4 x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(2 x \right)}}{x^{2} + 1}\right) - \frac{4 \log{\left(x^{2} + 1 \right)}}{\left(4 x^{2} + 1\right)^{2}} + \frac{3}{\left(x^{2} + 1\right) \left(4 x^{2} + 1\right)}\right)$$
Tercera derivada [src]
                    /                             3                                                                                                                                                                                                                                                \
          atan(2*x) |  /   /     2\              \         /     2\     /   /     2\              \ /                                       2                    /     2\\                                    2                      2                              3                 2    /     2\|
  /     2\          |  |log\1 + x /   x*atan(2*x)|    4*log\1 + x /     |log\1 + x /   x*atan(2*x)| |  atan(2*x)           4*x           2*x *atan(2*x)   8*x*log\1 + x /|            3                   24*x                    6*x            3*x*atan(2*x)   4*x *atan(2*x)   64*x *log\1 + x /|
4*\1 + x /         *|2*|----------- + -----------|  - ------------- - 3*|----------- + -----------|*|- --------- - ------------------- + -------------- + ---------------| + ------------------- - -------------------- - -------------------- - ------------- + -------------- + -----------------|
                    |  |         2            2  |               2      |         2            2  | |         2    /     2\ /       2\             2                  2  |   /     2\ /       2\                      2           2                        2               3                   3   |
                    |  \  1 + 4*x        1 + x   /     /       2\       \  1 + 4*x        1 + x   / |    1 + x     \1 + x /*\1 + 4*x /     /     2\         /       2\   |   \1 + x /*\1 + 4*x /   /     2\ /       2\    /     2\  /       2\     /     2\        /     2\          /       2\    |
                    \                                  \1 + 4*x /                                   \                                      \1 + x /         \1 + 4*x /   /                         \1 + x /*\1 + 4*x /    \1 + x / *\1 + 4*x /     \1 + x /        \1 + x /          \1 + 4*x /    /
$$4 \left(x^{2} + 1\right)^{\operatorname{atan}{\left(2 x \right)}} \left(\frac{4 x^{3} \operatorname{atan}{\left(2 x \right)}}{\left(x^{2} + 1\right)^{3}} + \frac{64 x^{2} \log{\left(x^{2} + 1 \right)}}{\left(4 x^{2} + 1\right)^{3}} - \frac{24 x^{2}}{\left(x^{2} + 1\right) \left(4 x^{2} + 1\right)^{2}} - \frac{6 x^{2}}{\left(x^{2} + 1\right)^{2} \left(4 x^{2} + 1\right)} - \frac{3 x \operatorname{atan}{\left(2 x \right)}}{\left(x^{2} + 1\right)^{2}} + 2 \left(\frac{x \operatorname{atan}{\left(2 x \right)}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{4 x^{2} + 1}\right)^{3} - 3 \left(\frac{x \operatorname{atan}{\left(2 x \right)}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{4 x^{2} + 1}\right) \left(\frac{2 x^{2} \operatorname{atan}{\left(2 x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{8 x \log{\left(x^{2} + 1 \right)}}{\left(4 x^{2} + 1\right)^{2}} - \frac{4 x}{\left(x^{2} + 1\right) \left(4 x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(2 x \right)}}{x^{2} + 1}\right) - \frac{4 \log{\left(x^{2} + 1 \right)}}{\left(4 x^{2} + 1\right)^{2}} + \frac{3}{\left(x^{2} + 1\right) \left(4 x^{2} + 1\right)}\right)$$
Gráfico
Derivada de y=(1+x^2)^arctg(2x)