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y=(cos(x/4))^(ln(x+3x))

Derivada de y=(cos(x/4))^(ln(x+3x))

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Solución

Ha introducido [src]
   log(x + 3*x)/x\
cos            |-|
               \4/
$$\cos^{\log{\left(x + 3 x \right)}}{\left(\frac{x}{4} \right)}$$
cos(x/4)^log(x + 3*x)
Solución detallada
  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada

  2. Simplificamos:


Respuesta:

Gráfica
Primera derivada [src]
                   /     /   /x\\                   /x\\
                   |4*log|cos|-||   log(x + 3*x)*sin|-||
   log(x + 3*x)/x\ |     \   \4//                   \4/|
cos            |-|*|------------- - -------------------|
               \4/ |   x + 3*x                 /x\     |
                   |                      4*cos|-|     |
                   \                           \4/     /
$$\left(- \frac{\log{\left(x + 3 x \right)} \sin{\left(\frac{x}{4} \right)}}{4 \cos{\left(\frac{x}{4} \right)}} + \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x + 3 x}\right) \cos^{\log{\left(x + 3 x \right)}}{\left(\frac{x}{4} \right)}$$
Segunda derivada [src]
               /                                                2                                              \
               |             /       /   /x\\               /x\\                                               |
               |             |  4*log|cos|-||   log(4*x)*sin|-||                                               |
               |             |       \   \4//               \4/|                                               |
               |             |- ------------- + ---------------|                                               |
               |             |        x                 /x\    |       /   /x\\        /x\        2/x\         |
               |             |                       cos|-|    |    log|cos|-||     sin|-|     sin |-|*log(4*x)|
   log(4*x)/x\ |  log(4*x)   \                          \4/    /       \   \4//        \4/         \4/         |
cos        |-|*|- -------- + ------------------------------------ - ----------- - ---------- - ----------------|
           \4/ |     16                       16                          2              /x\            2/x\   |
               |                                                         x        2*x*cos|-|      16*cos |-|   |
               \                                                                         \4/             \4/   /
$$\left(\frac{\left(\frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{\cos{\left(\frac{x}{4} \right)}} - \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x}\right)^{2}}{16} - \frac{\log{\left(4 x \right)} \sin^{2}{\left(\frac{x}{4} \right)}}{16 \cos^{2}{\left(\frac{x}{4} \right)}} - \frac{\log{\left(4 x \right)}}{16} - \frac{\sin{\left(\frac{x}{4} \right)}}{2 x \cos{\left(\frac{x}{4} \right)}} - \frac{\log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x^{2}}\right) \cos^{\log{\left(4 x \right)}}{\left(\frac{x}{4} \right)}$$
Tercera derivada [src]
               /                                            3                                                                                                                                                                                    \
               |         /       /   /x\\               /x\\                      /       /   /x\\               /x\\ /      /   /x\\      2/x\                 /x\           \                                                                  |
               |         |  4*log|cos|-||   log(4*x)*sin|-||                      |  4*log|cos|-||   log(4*x)*sin|-|| |16*log|cos|-||   sin |-|*log(4*x)   8*sin|-|           |                                                                  |
               |         |       \   \4//               \4/|                      |       \   \4//               \4/| |      \   \4//       \4/                 \4/           |                                                                  |
               |         |- ------------- + ---------------|                    3*|- ------------- + ---------------|*|-------------- + ---------------- + -------- + log(4*x)|                                                                  |
               |         |        x                 /x\    |         /   /x\\     |        x                 /x\    | |       2                2/x\             /x\           |         2/x\                 /x\      3/x\                   /x\ |
               |         |                       cos|-|    |    2*log|cos|-||     |                       cos|-|    | |      x              cos |-|        x*cos|-|           |    3*sin |-|     log(4*x)*sin|-|   sin |-|*log(4*x)     3*sin|-| |
   log(4*x)/x\ |   3     \                          \4/    /         \   \4//     \                          \4/    / \                         \4/             \4/           /          \4/                 \4/       \4/                   \4/ |
cos        |-|*|- ---- - ------------------------------------ + ------------- + ----------------------------------------------------------------------------------------------- - ------------ - --------------- - ---------------- + -----------|
           \4/ |  16*x                    64                           3                                                       64                                                         2/x\            /x\               3/x\         2    /x\|
               |                                                      x                                                                                                           16*x*cos |-|      32*cos|-|         32*cos |-|      4*x *cos|-||
               \                                                                                                                                                                           \4/            \4/                \4/              \4//
$$\left(- \frac{\left(\frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{\cos{\left(\frac{x}{4} \right)}} - \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x}\right)^{3}}{64} + \frac{3 \left(\frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{\cos{\left(\frac{x}{4} \right)}} - \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x}\right) \left(\frac{\log{\left(4 x \right)} \sin^{2}{\left(\frac{x}{4} \right)}}{\cos^{2}{\left(\frac{x}{4} \right)}} + \log{\left(4 x \right)} + \frac{8 \sin{\left(\frac{x}{4} \right)}}{x \cos{\left(\frac{x}{4} \right)}} + \frac{16 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x^{2}}\right)}{64} - \frac{\log{\left(4 x \right)} \sin^{3}{\left(\frac{x}{4} \right)}}{32 \cos^{3}{\left(\frac{x}{4} \right)}} - \frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{32 \cos{\left(\frac{x}{4} \right)}} - \frac{3 \sin^{2}{\left(\frac{x}{4} \right)}}{16 x \cos^{2}{\left(\frac{x}{4} \right)}} - \frac{3}{16 x} + \frac{3 \sin{\left(\frac{x}{4} \right)}}{4 x^{2} \cos{\left(\frac{x}{4} \right)}} + \frac{2 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x^{3}}\right) \cos^{\log{\left(4 x \right)}}{\left(\frac{x}{4} \right)}$$
Gráfico
Derivada de y=(cos(x/4))^(ln(x+3x))