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y=(4/x+√4x)^cos(4x)

Derivada de y=(4/x+√4x)^cos(4x)

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Solución

Ha introducido [src]
             cos(4*x)
/4     _____\        
|- + \/ 4*x |        
\x          /        
$$\left(\sqrt{4 x} + \frac{4}{x}\right)^{\cos{\left(4 x \right)}}$$
(4/x + sqrt(4*x))^cos(4*x)
Solución detallada
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Gráfica
Primera derivada [src]
                      /                                /           ___\         \
                      |                                |  4    2*\/ x |         |
                      |                                |- -- + -------|*cos(4*x)|
             cos(4*x) |                                |   2     2*x  |         |
/4     _____\         |       /4     _____\            \  x           /         |
|- + \/ 4*x |        *|- 4*log|- + \/ 4*x |*sin(4*x) + -------------------------|
\x          /         |       \x          /                   4     _____       |
                      |                                       - + \/ 4*x        |
                      \                                       x                 /
$$\left(\sqrt{4 x} + \frac{4}{x}\right)^{\cos{\left(4 x \right)}} \left(- 4 \log{\left(\sqrt{4 x} + \frac{4}{x} \right)} \sin{\left(4 x \right)} + \frac{\left(\frac{2 \sqrt{x}}{2 x} - \frac{4}{x^{2}}\right) \cos{\left(4 x \right)}}{\sqrt{4 x} + \frac{4}{x}}\right)$$
Segunda derivada [src]
                        /                                                       2                                                                                                             \
                        |/                                /  1     4 \         \                                                                                                              |
                        ||                                |----- - --|*cos(4*x)|                                                                                                              |
                        ||                                |  ___    2|         |                                                                                                              |
                        ||     /  /  ___   2\\            \\/ x    x /         |                                                                           2                                  |
                        ||8*log|2*|\/ x  + -||*sin(4*x) - ---------------------|                                       /  1     4 \            /  1     4 \             /   1     16\         |
                        ||     \  \        x//                    ___   2      |                                     4*|----- - --|*sin(4*x)   |----- - --| *cos(4*x)   |- ---- + --|*cos(4*x)|
               cos(4*x) ||                                      \/ x  + -      |                                       |  ___    2|            |  ___    2|             |   3/2    3|         |
/  /  ___   2\\         |\                                              x      /                   /  /  ___   2\\     \\/ x    x /            \\/ x    x /             \  x      x /         |
|2*|\/ x  + -||        *|-------------------------------------------------------- - 16*cos(4*x)*log|2*|\/ x  + -|| - ----------------------- - ---------------------- + ----------------------|
\  \        x//         |                           4                                              \  \        x//            ___   2                           2             /  ___   2\     |
                        |                                                                                                   \/ x  + -                /  ___   2\            4*|\/ x  + -|     |
                        |                                                                                                           x              4*|\/ x  + -|              \        x/     |
                        \                                                                                                                            \        x/                              /
$$\left(2 \left(\sqrt{x} + \frac{2}{x}\right)\right)^{\cos{\left(4 x \right)}} \left(\frac{\left(\frac{16}{x^{3}} - \frac{1}{x^{\frac{3}{2}}}\right) \cos{\left(4 x \right)}}{4 \left(\sqrt{x} + \frac{2}{x}\right)} - \frac{\left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right)^{2} \cos{\left(4 x \right)}}{4 \left(\sqrt{x} + \frac{2}{x}\right)^{2}} - \frac{4 \left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right) \sin{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} + \frac{\left(- \frac{\left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right) \cos{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} + 8 \log{\left(2 \left(\sqrt{x} + \frac{2}{x}\right) \right)} \sin{\left(4 x \right)}\right)^{2}}{4} - 16 \log{\left(2 \left(\sqrt{x} + \frac{2}{x}\right) \right)} \cos{\left(4 x \right)}\right)$$
Tercera derivada [src]
                        /                                                                                                                                                        /                                             2                                                             \                                                                                                                                                                             \
                        |                                                         3                                      /                                /  1     4 \         \ |                                 /  1     4 \             /   1     16\               /  1     4 \         |                                                                                                                                                                             |
                        |  /                                /  1     4 \         \                                       |                                |----- - --|*cos(4*x)| |                                 |----- - --| *cos(4*x)   |- ---- + --|*cos(4*x)   16*|----- - --|*sin(4*x)|                                                                                                                                                                             |
                        |  |                                |----- - --|*cos(4*x)|                                       |                                |  ___    2|         | |                                 |  ___    2|             |   3/2    3|               |  ___    2|         |                                                                                                                                                                             |
                        |  |                                |  ___    2|         |                                       |     /  /  ___   2\\            \\/ x    x /         | |               /  /  ___   2\\   \\/ x    x /             \  x      x /               \\/ x    x /         |                                                                                                                                                                             |
                        |  |     /  /  ___   2\\            \\/ x    x /         |                                     3*|8*log|2*|\/ x  + -||*sin(4*x) - ---------------------|*|64*cos(4*x)*log|2*|\/ x  + -|| + ---------------------- - ---------------------- + ------------------------|                                                                       2                                                   3                                                 |
                        |  |8*log|2*|\/ x  + -||*sin(4*x) - ---------------------|                                       |     \  \        x//                    ___   2      | |               \  \        x//                   2                ___   2                   ___   2        |      /  1     4 \              /   1     16\              /  1     4 \               /   1     32\            /  1     4 \               /  1     4 \ /   1     16\         |
                        |  |     \  \        x//                    ___   2      |                                       |                                      \/ x  + -      | |                                      /  ___   2\               \/ x  + -                 \/ x  + -        |   24*|----- - --|*cos(4*x)   3*|- ---- + --|*sin(4*x)   3*|----- - --| *sin(4*x)   3*|- ---- + --|*cos(4*x)   |----- - --| *cos(4*x)   3*|----- - --|*|- ---- + --|*cos(4*x)|
               cos(4*x) |  |                                      \/ x  + -      |                                       \                                              x      / |                                      |\/ x  + -|                       x                         x        |      |  ___    2|              |   3/2    3|              |  ___    2|               |   5/2    4|            |  ___    2|               |  ___    2| |   3/2    3|         |
/  /  ___   2\\         |  \                                              x      /          /  /  ___   2\\                                                                      \                                      \        x/                                                          /      \\/ x    x /              \  x      x /              \\/ x    x /               \  x      x /            \\/ x    x /               \\/ x    x / \  x      x /         |
|2*|\/ x  + -||        *|- -------------------------------------------------------- + 64*log|2*|\/ x  + -||*sin(4*x) + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------ - ------------------------ + ------------------------ - ------------------------ + ---------------------- - -------------------------------------|
\  \        x//         |                             8                                     \  \        x//                                                                                               8                                                                                               ___   2                    ___   2                            2                /  ___   2\                          3                                2           |
                        |                                                                                                                                                                                                                                                                               \/ x  + -                  \/ x  + -                 /  ___   2\               8*|\/ x  + -|               /  ___   2\                      /  ___   2\            |
                        |                                                                                                                                                                                                                                                                                       x                          x                 |\/ x  + -|                 \        x/             4*|\/ x  + -|                    8*|\/ x  + -|            |
                        \                                                                                                                                                                                                                                                                                                                                    \        x/                                           \        x/                      \        x/            /
$$\left(2 \left(\sqrt{x} + \frac{2}{x}\right)\right)^{\cos{\left(4 x \right)}} \left(- \frac{3 \left(\frac{32}{x^{4}} - \frac{1}{x^{\frac{5}{2}}}\right) \cos{\left(4 x \right)}}{8 \left(\sqrt{x} + \frac{2}{x}\right)} - \frac{3 \left(\frac{16}{x^{3}} - \frac{1}{x^{\frac{3}{2}}}\right) \left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right) \cos{\left(4 x \right)}}{8 \left(\sqrt{x} + \frac{2}{x}\right)^{2}} - \frac{3 \left(\frac{16}{x^{3}} - \frac{1}{x^{\frac{3}{2}}}\right) \sin{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} + \frac{\left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right)^{3} \cos{\left(4 x \right)}}{4 \left(\sqrt{x} + \frac{2}{x}\right)^{3}} + \frac{3 \left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right)^{2} \sin{\left(4 x \right)}}{\left(\sqrt{x} + \frac{2}{x}\right)^{2}} - \frac{24 \left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right) \cos{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} - \frac{\left(- \frac{\left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right) \cos{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} + 8 \log{\left(2 \left(\sqrt{x} + \frac{2}{x}\right) \right)} \sin{\left(4 x \right)}\right)^{3}}{8} + \frac{3 \left(- \frac{\left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right) \cos{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} + 8 \log{\left(2 \left(\sqrt{x} + \frac{2}{x}\right) \right)} \sin{\left(4 x \right)}\right) \left(- \frac{\left(\frac{16}{x^{3}} - \frac{1}{x^{\frac{3}{2}}}\right) \cos{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} + \frac{\left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right)^{2} \cos{\left(4 x \right)}}{\left(\sqrt{x} + \frac{2}{x}\right)^{2}} + \frac{16 \left(- \frac{4}{x^{2}} + \frac{1}{\sqrt{x}}\right) \sin{\left(4 x \right)}}{\sqrt{x} + \frac{2}{x}} + 64 \log{\left(2 \left(\sqrt{x} + \frac{2}{x}\right) \right)} \cos{\left(4 x \right)}\right)}{8} + 64 \log{\left(2 \left(\sqrt{x} + \frac{2}{x}\right) \right)} \sin{\left(4 x \right)}\right)$$
Gráfico
Derivada de y=(4/x+√4x)^cos(4x)