Derivada de y=(sin5x)^x

Ha introducido [src]

   x     
sin (5*x)

$$\sin^{x}{\left(5 x \right)}$$

sin(5*x)^x

Solución detallada

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Perola derivada

$x^{x} \left(\log{\left(x \right)} + 1\right)$

Respuesta:

$x^{x} \left(\log{\left(x \right)} + 1\right)$

Gráfica

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Primera derivada [src]

   x      /5*x*cos(5*x)                \
sin (5*x)*|------------ + log(sin(5*x))|
          \  sin(5*x)                  /

$$\left(\frac{5 x \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}} + \log{\left(\sin{\left(5 x \right)} \right)}\right) \sin^{x}{\left(5 x \right)}$$

Simplificar

Segunda derivada [src]

          /                              2                                2     \
   x      |/5*x*cos(5*x)                \           10*cos(5*x)   25*x*cos (5*x)|
sin (5*x)*||------------ + log(sin(5*x))|  - 25*x + ----------- - --------------|
          |\  sin(5*x)                  /             sin(5*x)         2        |
          \                                                         sin (5*x)   /

$$\left(- 25 x - \frac{25 x \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}} + \left(\frac{5 x \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}} + \log{\left(\sin{\left(5 x \right)} \right)}\right)^{2} + \frac{10 \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}\right) \sin^{x}{\left(5 x \right)}$$

Simplificar

Tercera derivada [src]

          /                                    3         2                                          /                          2     \            3                      \
   x      |      /5*x*cos(5*x)                \    75*cos (5*x)      /5*x*cos(5*x)                \ |      2*cos(5*x)   5*x*cos (5*x)|   250*x*cos (5*x)   250*x*cos(5*x)|
sin (5*x)*|-75 + |------------ + log(sin(5*x))|  - ------------ - 15*|------------ + log(sin(5*x))|*|5*x - ---------- + -------------| + --------------- + --------------|
          |      \  sin(5*x)                  /        2             \  sin(5*x)                  / |       sin(5*x)         2       |         3              sin(5*x)   |
          \                                         sin (5*x)                                       \                     sin (5*x)  /      sin (5*x)                    /

$$\left(\frac{250 x \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}} + \frac{250 x \cos^{3}{\left(5 x \right)}}{\sin^{3}{\left(5 x \right)}} + \left(\frac{5 x \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}} + \log{\left(\sin{\left(5 x \right)} \right)}\right)^{3} - 15 \left(\frac{5 x \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}} + \log{\left(\sin{\left(5 x \right)} \right)}\right) \left(5 x + \frac{5 x \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}} - \frac{2 \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}\right) - 75 - \frac{75 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \sin^{x}{\left(5 x \right)}$$

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Gráfico

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Derivada de y=(sin5x)^x

Gráfico:

Definida a trozos:

Solución