Derivada de y=cosx^✓x

Ha introducido [src]

          ___
        \/ x 
(cos(x))

$$\cos^{\sqrt{x}}{\left(x \right)}$$

cos(x)^(sqrt(x))

Solución detallada

No logro encontrar los pasos en la búsqueda de esta derivada.

Perola derivada

$x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right)$
Simplificamos:

$\frac{x^{\frac{\sqrt{x}}{2}} \left(\log{\left(x \right)} + 2\right)}{2}$

Respuesta:

$\frac{x^{\frac{\sqrt{x}}{2}} \left(\log{\left(x \right)} + 2\right)}{2}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

          ___ /                ___       \
        \/ x  |log(cos(x))   \/ x *sin(x)|
(cos(x))     *|----------- - ------------|
              |      ___        cos(x)   |
              \  2*\/ x                  /

$$\left(- \frac{\sqrt{x} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{2 \sqrt{x}}\right) \cos^{\sqrt{x}}{\left(x \right)}$$

Simplificar

Segunda derivada [src]

              /                                          2                                             \
              |          /                    ___       \                                              |
              |          |  log(cos(x))   2*\/ x *sin(x)|                                              |
              |          |- ----------- + --------------|                                              |
          ___ |          |       ___          cos(x)    |                    ___    2                  |
        \/ x  |    ___   \     \/ x                     /    log(cos(x))   \/ x *sin (x)      sin(x)   |
(cos(x))     *|- \/ x  + --------------------------------- - ----------- - ------------- - ------------|
              |                          4                         3/2           2           ___       |
              \                                                 4*x           cos (x)      \/ x *cos(x)/

$$\left(- \frac{\sqrt{x} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \sqrt{x} + \frac{\left(\frac{2 \sqrt{x} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{\sqrt{x}}\right)^{2}}{4} - \frac{\sin{\left(x \right)}}{\sqrt{x} \cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{4 x^{\frac{3}{2}}}\right) \cos^{\sqrt{x}}{\left(x \right)}$$

Simplificar

Tercera derivada [src]

              /                                            3                                                                                                                                                                                   \
              |            /                    ___       \                      /                    ___       \ /                            ___    2                  \                                                                     |
              |            |  log(cos(x))   2*\/ x *sin(x)|                      |  log(cos(x))   2*\/ x *sin(x)| |    ___   log(cos(x))   4*\/ x *sin (x)     4*sin(x)  |                                                                     |
              |            |- ----------- + --------------|                    3*|- ----------- + --------------|*|4*\/ x  + ----------- + --------------- + ------------|                                                                     |
          ___ |            |       ___          cos(x)    |                      |       ___          cos(x)    | |               3/2             2            ___       |       ___              ___    3              2                      |
        \/ x  |     3      \     \/ x                     /    3*log(cos(x))     \     \/ x                     / \              x             cos (x)       \/ x *cos(x)/   2*\/ x *sin(x)   2*\/ x *sin (x)      3*sin (x)         3*sin(x)  |
(cos(x))     *|- ------- - --------------------------------- + ------------- + ------------------------------------------------------------------------------------------- - -------------- - --------------- - --------------- + -------------|
              |      ___                   8                          5/2                                                   8                                                    cos(x)              3              ___    2         3/2       |
              \  2*\/ x                                            8*x                                                                                                                            cos (x)       2*\/ x *cos (x)   4*x   *cos(x)/

$$\left(- \frac{2 \sqrt{x} \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \sqrt{x} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\left(\frac{2 \sqrt{x} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{\sqrt{x}}\right)^{3}}{8} + \frac{3 \left(\frac{2 \sqrt{x} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{\sqrt{x}}\right) \left(\frac{4 \sqrt{x} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 4 \sqrt{x} + \frac{4 \sin{\left(x \right)}}{\sqrt{x} \cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{\frac{3}{2}}}\right)}{8} - \frac{3 \sin^{2}{\left(x \right)}}{2 \sqrt{x} \cos^{2}{\left(x \right)}} - \frac{3}{2 \sqrt{x}} + \frac{3 \sin{\left(x \right)}}{4 x^{\frac{3}{2}} \cos{\left(x \right)}} + \frac{3 \log{\left(\cos{\left(x \right)} \right)}}{8 x^{\frac{5}{2}}}\right) \cos^{\sqrt{x}}{\left(x \right)}$$

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Gráfico

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Expresiones idénticas

Derivada de y=cosx^✓x

Gráfico:

Definida a trozos:

Solución