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Derivada de y=(1+5^x^2)^x

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

Ha introducido [src]
           x
/     / 2\\ 
|     \x /| 
\1 + 5    / 
$$\left(5^{x^{2}} + 1\right)^{x}$$
(1 + 5^(x^2))^x
Solución detallada
  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada


Respuesta:

Primera derivada [src]
           x /   / 2\                           \
/     / 2\\  |   \x /  2             /     / 2\\|
|     \x /|  |2*5    *x *log(5)      |     \x /||
\1 + 5    / *|----------------- + log\1 + 5    /|
             |         / 2\                     |
             |         \x /                     |
             \    1 + 5                         /
$$\left(5^{x^{2}} + 1\right)^{x} \left(\frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + \log{\left(5^{x^{2}} + 1 \right)}\right)$$
Segunda derivada [src]
             /                                                  /                     / 2\          \       \
             |                                             / 2\ |                     \x /  2       |       |
             |                                             \x / |       2          2*5    *x *log(5)|       |
             |                                    2   2*x*5    *|3 + 2*x *log(5) - -----------------|*log(5)|
           x |/   / 2\                           \              |                           / 2\    |       |
/     / 2\\  ||   \x /  2             /     / 2\\|              |                           \x /    |       |
|     \x /|  ||2*5    *x *log(5)      |     \x /||              \                      1 + 5        /       |
\1 + 5    / *||----------------- + log\1 + 5    /|  + ------------------------------------------------------|
             ||         / 2\                     |                               / 2\                       |
             ||         \x /                     |                               \x /                       |
             \\    1 + 5                         /                          1 + 5                           /
$$\left(5^{x^{2}} + 1\right)^{x} \left(\frac{2 \cdot 5^{x^{2}} x \left(- \frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + 2 x^{2} \log{\left(5 \right)} + 3\right) \log{\left(5 \right)}}{5^{x^{2}} + 1} + \left(\frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + \log{\left(5^{x^{2}} + 1 \right)}\right)^{2}\right)$$
3-я производная [src]
             /                                                /                                      / 2\                 / 2\                    2           \                                                                                                     \
             |                                           / 2\ |                                      \x /  2              \x /  4    2         2*x   4    2   |                    /   / 2\                           \ /                     / 2\          \       |
             |                                           \x / |       4    2          2          12*5    *x *log(5)   12*5    *x *log (5)   8*5    *x *log (5)|               / 2\ |   \x /  2             /     / 2\\| |                     \x /  2       |       |
             |                                        2*5    *|3 + 4*x *log (5) + 12*x *log(5) - ------------------ - ------------------- + ------------------|*log(5)        \x / |2*5    *x *log(5)      |     \x /|| |       2          2*5    *x *log(5)|       |
             |                                    3           |                                           / 2\                  / 2\                      2   |          6*x*5    *|----------------- + log\1 + 5    /|*|3 + 2*x *log(5) - -----------------|*log(5)|
           x |/   / 2\                           \            |                                           \x /                  \x /           /     / 2\\    |                    |         / 2\                     | |                           / 2\    |       |
/     / 2\\  ||   \x /  2             /     / 2\\|            |                                      1 + 5                 1 + 5               |     \x /|    |                    |         \x /                     | |                           \x /    |       |
|     \x /|  ||2*5    *x *log(5)      |     \x /||            \                                                                                \1 + 5    /    /                    \    1 + 5                         / \                      1 + 5        /       |
\1 + 5    / *||----------------- + log\1 + 5    /|  + ---------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------|
             ||         / 2\                     |                                                            / 2\                                                                                                     / 2\                                         |
             ||         \x /                     |                                                            \x /                                                                                                     \x /                                         |
             \\    1 + 5                         /                                                       1 + 5                                                                                                    1 + 5                                             /
$$\left(5^{x^{2}} + 1\right)^{x} \left(\frac{6 \cdot 5^{x^{2}} x \left(\frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + \log{\left(5^{x^{2}} + 1 \right)}\right) \left(- \frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + 2 x^{2} \log{\left(5 \right)} + 3\right) \log{\left(5 \right)}}{5^{x^{2}} + 1} + \frac{2 \cdot 5^{x^{2}} \left(\frac{8 \cdot 5^{2 x^{2}} x^{4} \log{\left(5 \right)}^{2}}{\left(5^{x^{2}} + 1\right)^{2}} - \frac{12 \cdot 5^{x^{2}} x^{4} \log{\left(5 \right)}^{2}}{5^{x^{2}} + 1} - \frac{12 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + 4 x^{4} \log{\left(5 \right)}^{2} + 12 x^{2} \log{\left(5 \right)} + 3\right) \log{\left(5 \right)}}{5^{x^{2}} + 1} + \left(\frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + \log{\left(5^{x^{2}} + 1 \right)}\right)^{3}\right)$$
Tercera derivada [src]
             /                                                /                                      / 2\                 / 2\                    2           \                                                                                                     \
             |                                           / 2\ |                                      \x /  2              \x /  4    2         2*x   4    2   |                    /   / 2\                           \ /                     / 2\          \       |
             |                                           \x / |       4    2          2          12*5    *x *log(5)   12*5    *x *log (5)   8*5    *x *log (5)|               / 2\ |   \x /  2             /     / 2\\| |                     \x /  2       |       |
             |                                        2*5    *|3 + 4*x *log (5) + 12*x *log(5) - ------------------ - ------------------- + ------------------|*log(5)        \x / |2*5    *x *log(5)      |     \x /|| |       2          2*5    *x *log(5)|       |
             |                                    3           |                                           / 2\                  / 2\                      2   |          6*x*5    *|----------------- + log\1 + 5    /|*|3 + 2*x *log(5) - -----------------|*log(5)|
           x |/   / 2\                           \            |                                           \x /                  \x /           /     / 2\\    |                    |         / 2\                     | |                           / 2\    |       |
/     / 2\\  ||   \x /  2             /     / 2\\|            |                                      1 + 5                 1 + 5               |     \x /|    |                    |         \x /                     | |                           \x /    |       |
|     \x /|  ||2*5    *x *log(5)      |     \x /||            \                                                                                \1 + 5    /    /                    \    1 + 5                         / \                      1 + 5        /       |
\1 + 5    / *||----------------- + log\1 + 5    /|  + ---------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------|
             ||         / 2\                     |                                                            / 2\                                                                                                     / 2\                                         |
             ||         \x /                     |                                                            \x /                                                                                                     \x /                                         |
             \\    1 + 5                         /                                                       1 + 5                                                                                                    1 + 5                                             /
$$\left(5^{x^{2}} + 1\right)^{x} \left(\frac{6 \cdot 5^{x^{2}} x \left(\frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + \log{\left(5^{x^{2}} + 1 \right)}\right) \left(- \frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + 2 x^{2} \log{\left(5 \right)} + 3\right) \log{\left(5 \right)}}{5^{x^{2}} + 1} + \frac{2 \cdot 5^{x^{2}} \left(\frac{8 \cdot 5^{2 x^{2}} x^{4} \log{\left(5 \right)}^{2}}{\left(5^{x^{2}} + 1\right)^{2}} - \frac{12 \cdot 5^{x^{2}} x^{4} \log{\left(5 \right)}^{2}}{5^{x^{2}} + 1} - \frac{12 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + 4 x^{4} \log{\left(5 \right)}^{2} + 12 x^{2} \log{\left(5 \right)} + 3\right) \log{\left(5 \right)}}{5^{x^{2}} + 1} + \left(\frac{2 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}}{5^{x^{2}} + 1} + \log{\left(5^{x^{2}} + 1 \right)}\right)^{3}\right)$$