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y=ln(4^x+5^x^2)^(1/x)

Derivada de y=ln(4^x+5^x^2)^(1/x)

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
    _________________
   /    /      / 2\\ 
x /     | x    \x /| 
\/   log\4  + 5    / 
$$\log{\left(4^{x} + 5^{x^{2}} \right)}^{\frac{1}{x}}$$
log(4^x + 5^(x^2))^(1/x)
Solución detallada
  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada


Respuesta:

Gráfica
Primera derivada [src]
    _________________ /     /   /      / 2\\\                     / 2\        \
   /    /      / 2\\  |     |   | x    \x /||     x               \x /        |
x /     | x    \x /|  |  log\log\4  + 5    //    4 *log(4) + 2*x*5    *log(5) |
\/   log\4  + 5    / *|- -------------------- + ------------------------------|
                      |            2              /      / 2\\    /      / 2\\|
                      |           x               | x    \x /|    | x    \x /||
                      \                         x*\4  + 5    /*log\4  + 5    //
$$\left(\frac{4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}}{x \left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{\log{\left(\log{\left(4^{x} + 5^{x^{2}} \right)} \right)}}{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}^{\frac{1}{x}}$$
Segunda derivada [src]
                      /                                                       2                                                                                                                                                                                   \
                      |/     /   /      / 2\\\                    / 2\       \                                                                                                                                                                                    |
                      ||     |   | x    \x /||    x               \x /       |                                                                                                                                                                                    |
                      ||  log\log\4  + 5    //   4 *log(4) + 2*x*5    *log(5)|                                                                                                                                                                                    |
                      ||- -------------------- + ----------------------------|                                                                                                              2                                 2                                   |
    _________________ ||           x             /      / 2\\    /      / 2\\|         /   /      / 2\\\                   / 2\             / 2\              /                 / 2\       \    /                 / 2\       \      /                 / 2\       \|
   /    /      / 2\\  ||                         | x    \x /|    | x    \x /||         |   | x    \x /||    x    2         \x /             \x /  2    2      | x               \x /       |    | x               \x /       |      | x               \x /       ||
x /     | x    \x /|  |\                         \4  + 5    /*log\4  + 5    //    2*log\log\4  + 5    //   4 *log (4) + 2*5    *log(5) + 4*5    *x *log (5)   \4 *log(4) + 2*x*5    *log(5)/    \4 *log(4) + 2*x*5    *log(5)/    2*\4 *log(4) + 2*x*5    *log(5)/|
\/   log\4  + 5    / *|-------------------------------------------------------- + ---------------------- + ------------------------------------------------ - ------------------------------- - ------------------------------- - --------------------------------|
                      |                           x                                          2                       /      / 2\\    /      / 2\\                          2                                 2                       /      / 2\\    /      / 2\\ |
                      |                                                                     x                        | x    \x /|    | x    \x /|              /      / 2\\     /      / 2\\     /      / 2\\      /      / 2\\      | x    \x /|    | x    \x /| |
                      |                                                                                              \4  + 5    /*log\4  + 5    /              | x    \x /|     | x    \x /|     | x    \x /|     2| x    \x /|    x*\4  + 5    /*log\4  + 5    / |
                      \                                                                                                                                        \4  + 5    / *log\4  + 5    /     \4  + 5    / *log \4  + 5    /                                   /
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                                                                                                                                 x                                                                                                                                 
$$\frac{\left(\frac{4^{x} \log{\left(4 \right)}^{2} + 4 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}^{2} + 2 \cdot 5^{x^{2}} \log{\left(5 \right)}}{\left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{\left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{2}}{\left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{\left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{2}}{\left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}^{2}} + \frac{\left(\frac{4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}}{\left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{\log{\left(\log{\left(4^{x} + 5^{x^{2}} \right)} \right)}}{x}\right)^{2}}{x} - \frac{2 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)}{x \left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} + \frac{2 \log{\left(\log{\left(4^{x} + 5^{x^{2}} \right)} \right)}}{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}^{\frac{1}{x}}}{x}$$
Tercera derivada [src]
                      /                                                                                                                                                                                                     /                                                         2                                 2                                                                                      \                                                                                                                                                                                                                                                                                                                                                                                                                                                          \
                      |                                                       3                                                                                     /     /   /      / 2\\\                    / 2\       \ |       /   /      / 2\\\   /                 / 2\       \    /                 / 2\       \                    / 2\             / 2\                /                 / 2\       \|                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
                      |/     /   /      / 2\\\                    / 2\       \                                                                                      |     |   | x    \x /||    x               \x /       | |       |   | x    \x /||   | x               \x /       |    | x               \x /       |     x    2         \x /             \x /  2    2        | x               \x /       ||                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
                      ||     |   | x    \x /||    x               \x /       |                                                                                      |  log\log\4  + 5    //   4 *log(4) + 2*x*5    *log(5)| |  2*log\log\4  + 5    //   \4 *log(4) + 2*x*5    *log(5)/    \4 *log(4) + 2*x*5    *log(5)/    4 *log (4) + 2*5    *log(5) + 4*5    *x *log (5)   2*\4 *log(4) + 2*x*5    *log(5)/|                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
                      ||  log\log\4  + 5    //   4 *log(4) + 2*x*5    *log(5)|                                                                                    3*|- -------------------- + ----------------------------|*|- ---------------------- + ------------------------------- + ------------------------------- - ------------------------------------------------ + --------------------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
                      ||- -------------------- + ----------------------------|                                                                                      |           x             /      / 2\\    /      / 2\\| |             2                          2                                 2                              /      / 2\\    /      / 2\\                /      / 2\\    /      / 2\\ |                                   3                                   3                                   3                                                                                                                                                                                                                                                                      2                                   2                                   |
    _________________ ||           x             /      / 2\\    /      / 2\\|         /   /      / 2\\\                   / 2\                    / 2\             |                         | x    \x /|    | x    \x /|| |            x               /      / 2\\     /      / 2\\     /      / 2\\      /      / 2\\             | x    \x /|    | x    \x /|                | x    \x /|    | x    \x /| |     /                 / 2\       \      /                 / 2\       \      /                 / 2\       \      /                / 2\             / 2\           \     /                 / 2\       \ /                / 2\             / 2\           \     /                 / 2\       \ /                / 2\             / 2\           \     /                 / 2\       \      /                 / 2\       \      /                 / 2\       \|
   /    /      / 2\\  ||                         | x    \x /|    | x    \x /||         |   | x    \x /||    x    3         \x /  3    3            \x /    2        \                         \4  + 5    /*log\4  + 5    // |                            | x    \x /|     | x    \x /|     | x    \x /|     2| x    \x /|             \4  + 5    /*log\4  + 5    /              x*\4  + 5    /*log\4  + 5    / |     | x               \x /       |      | x               \x /       |      | x               \x /       |      | x    2         \x /             \x /  2    2   |     | x               \x /       | | x    2         \x /             \x /  2    2   |     | x               \x /       | | x    2         \x /             \x /  2    2   |     | x               \x /       |      | x               \x /       |      | x               \x /       ||
x /     | x    \x /|  |\                         \4  + 5    /*log\4  + 5    //    6*log\log\4  + 5    //   4 *log (4) + 8*5    *x *log (5) + 12*x*5    *log (5)                                                             \                            \4  + 5    / *log\4  + 5    /     \4  + 5    / *log \4  + 5    /                                                                                      /   2*\4 *log(4) + 2*x*5    *log(5)/    2*\4 *log(4) + 2*x*5    *log(5)/    3*\4 *log(4) + 2*x*5    *log(5)/    3*\4 *log (4) + 2*5    *log(5) + 4*5    *x *log (5)/   3*\4 *log(4) + 2*x*5    *log(5)/*\4 *log (4) + 2*5    *log(5) + 4*5    *x *log (5)/   3*\4 *log(4) + 2*x*5    *log(5)/*\4 *log (4) + 2*5    *log(5) + 4*5    *x *log (5)/   3*\4 *log(4) + 2*x*5    *log(5)/    3*\4 *log(4) + 2*x*5    *log(5)/    6*\4 *log(4) + 2*x*5    *log(5)/|
\/   log\4  + 5    / *|-------------------------------------------------------- - ---------------------- + ---------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------- + --------------------------------- + --------------------------------- - ---------------------------------------------------- - ----------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------- + --------------------------------- + --------------------------------- + --------------------------------|
                      |                            2                                         3                         /      / 2\\    /      / 2\\                                                                                                                                     x                                                                                                                                        3                                   3                                   3                                  /      / 2\\    /      / 2\\                                                     2                                                                                     2                                                             2                                   2                       /      / 2\\    /      / 2\\ |
                      |                           x                                         x                          | x    \x /|    | x    \x /|                                                                                                                                                                                                                                                                  /      / 2\\     /      / 2\\       /      / 2\\      /      / 2\\      /      / 2\\      /      / 2\\                 | x    \x /|    | x    \x /|                                         /      / 2\\     /      / 2\\                                                         /      / 2\\      /      / 2\\                                /      / 2\\     /      / 2\\       /      / 2\\      /      / 2\\    2 | x    \x /|    | x    \x /| |
                      |                                                                                                \4  + 5    /*log\4  + 5    /                                                                                                                                                                                                                                                                  | x    \x /|     | x    \x /|       | x    \x /|     3| x    \x /|      | x    \x /|     2| x    \x /|               x*\4  + 5    /*log\4  + 5    /                                         | x    \x /|     | x    \x /|                                                         | x    \x /|     2| x    \x /|                                | x    \x /|     | x    \x /|       | x    \x /|     2| x    \x /|   x *\4  + 5    /*log\4  + 5    / |
                      \                                                                                                                                                                                                                                                                                                                                                                                              \4  + 5    / *log\4  + 5    /       \4  + 5    / *log \4  + 5    /      \4  + 5    / *log \4  + 5    /                                                                                      \4  + 5    / *log\4  + 5    /                                                         \4  + 5    / *log \4  + 5    /                              x*\4  + 5    / *log\4  + 5    /     x*\4  + 5    / *log \4  + 5    /                                   /
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                                                                                                                                                                                                                                                                                                                                                                                                                                     x                                                                                                                                                                                                                                                                                                                                                                                                                                     
$$\frac{\left(\frac{4^{x} \log{\left(4 \right)}^{3} + 8 \cdot 5^{x^{2}} x^{3} \log{\left(5 \right)}^{3} + 12 \cdot 5^{x^{2}} x \log{\left(5 \right)}^{2}}{\left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{3 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right) \left(4^{x} \log{\left(4 \right)}^{2} + 4 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}^{2} + 2 \cdot 5^{x^{2}} \log{\left(5 \right)}\right)}{\left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{3 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right) \left(4^{x} \log{\left(4 \right)}^{2} + 4 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}^{2} + 2 \cdot 5^{x^{2}} \log{\left(5 \right)}\right)}{\left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}^{2}} + \frac{2 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{3}}{\left(4^{x} + 5^{x^{2}}\right)^{3} \log{\left(4^{x} + 5^{x^{2}} \right)}} + \frac{3 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{3}}{\left(4^{x} + 5^{x^{2}}\right)^{3} \log{\left(4^{x} + 5^{x^{2}} \right)}^{2}} + \frac{2 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{3}}{\left(4^{x} + 5^{x^{2}}\right)^{3} \log{\left(4^{x} + 5^{x^{2}} \right)}^{3}} - \frac{3 \left(\frac{4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}}{\left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{\log{\left(\log{\left(4^{x} + 5^{x^{2}} \right)} \right)}}{x}\right) \left(- \frac{4^{x} \log{\left(4 \right)}^{2} + 4 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}^{2} + 2 \cdot 5^{x^{2}} \log{\left(5 \right)}}{\left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} + \frac{\left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{2}}{\left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}} + \frac{\left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{2}}{\left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}^{2}} + \frac{2 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)}{x \left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{2 \log{\left(\log{\left(4^{x} + 5^{x^{2}} \right)} \right)}}{x^{2}}\right)}{x} - \frac{3 \left(4^{x} \log{\left(4 \right)}^{2} + 4 \cdot 5^{x^{2}} x^{2} \log{\left(5 \right)}^{2} + 2 \cdot 5^{x^{2}} \log{\left(5 \right)}\right)}{x \left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} + \frac{3 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{2}}{x \left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}} + \frac{3 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)^{2}}{x \left(4^{x} + 5^{x^{2}}\right)^{2} \log{\left(4^{x} + 5^{x^{2}} \right)}^{2}} + \frac{\left(\frac{4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}}{\left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{\log{\left(\log{\left(4^{x} + 5^{x^{2}} \right)} \right)}}{x}\right)^{3}}{x^{2}} + \frac{6 \left(4^{x} \log{\left(4 \right)} + 2 \cdot 5^{x^{2}} x \log{\left(5 \right)}\right)}{x^{2} \left(4^{x} + 5^{x^{2}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}} - \frac{6 \log{\left(\log{\left(4^{x} + 5^{x^{2}} \right)} \right)}}{x^{3}}\right) \log{\left(4^{x} + 5^{x^{2}} \right)}^{\frac{1}{x}}}{x}$$
Gráfico
Derivada de y=ln(4^x+5^x^2)^(1/x)