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Derivada de la función/ i*n*x/ x*sin/ sin(x)/ (x*sin(x))^(i*n*x*sin(x))

Derivada de (x*sin(x))^(i*n*x*sin(x))

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
          I*n*x*sin(x)
(x*sin(x))
(xsin⁡(x))xinsin⁡(x)\left(x \sin{\left(x \right)}\right)^{x i n \sin{\left(x \right)}} 
(x*sin(x))^(((i*n)*x)*sin(x))
Solución detallada

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

    Perola derivada

    (xinsin⁡(x))xinsin⁡(x)(log⁡(xinsin⁡(x))+1)\left(x i n \sin{\left(x \right)}\right)^{x i n \sin{\left(x \right)}} \left(\log{\left(x i n \sin{\left(x \right)} \right)} + 1\right)

  2. Simplificamos:

    (inxsin⁡(x))inxsin⁡(x)(log⁡(inxsin⁡(x))+1)\left(i n x \sin{\left(x \right)}\right)^{i n x \sin{\left(x \right)}} \left(\log{\left(i n x \sin{\left(x \right)} \right)} + 1\right)

Respuesta:

(inxsin⁡(x))inxsin⁡(x)(log⁡(inxsin⁡(x))+1)\left(i n x \sin{\left(x \right)}\right)^{i n x \sin{\left(x \right)}} \left(\log{\left(i n x \sin{\left(x \right)} \right)} + 1\right)

Primera derivada [src] 
          I*n*x*sin(x)                                                                      
(x*sin(x))            *((I*n*sin(x) + I*n*x*cos(x))*log(x*sin(x)) + I*n*(x*cos(x) + sin(x)))
(xsin⁡(x))xinsin⁡(x)(in(xcos⁡(x)+sin⁡(x))+(inxcos⁡(x)+insin⁡(x))log⁡(xsin⁡(x)))\left(x \sin{\left(x \right)}\right)^{x i n \sin{\left(x \right)}} \left(i n \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) + \left(i n x \cos{\left(x \right)} + i n \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)}\right)
Simplificar
Segunda derivada [src] 
                          /  /                                                                                 2\                                              \
             I*n*x*sin(x) |  |                                                              (x*cos(x) + sin(x)) |                        2                    2|
-n*(x*sin(x))            *|I*|-2*cos(x) + x*sin(x) + (-2*cos(x) + x*sin(x))*log(x*sin(x)) - --------------------| + n*(1 + log(x*sin(x))) *(x*cos(x) + sin(x)) |
                          \  \                                                                    x*sin(x)      /                                              /
−n(xsin⁡(x))inxsin⁡(x)(n(xcos⁡(x)+sin⁡(x))2(log⁡(xsin⁡(x))+1)2+i(xsin⁡(x)+(xsin⁡(x)−2cos⁡(x))log⁡(xsin⁡(x))−2cos⁡(x)−(xcos⁡(x)+sin⁡(x))2xsin⁡(x)))- n \left(x \sin{\left(x \right)}\right)^{i n x \sin{\left(x \right)}} \left(n \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \left(\log{\left(x \sin{\left(x \right)} \right)} + 1\right)^{2} + i \left(x \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} - 2 \cos{\left(x \right)} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2}}{x \sin{\left(x \right)}}\right)\right)
Simplificar
Tercera derivada [src] 
                         /    /                                                                               2                      2                                                      \                                                                                                /                                                                                 2\\
            I*n*x*sin(x) |    |                                                            (x*cos(x) + sin(x))    (x*cos(x) + sin(x)) *cos(x)   3*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))|      2                    3                    3                                               |                                                              (x*cos(x) + sin(x)) ||
n*(x*sin(x))            *|- I*|3*sin(x) + x*cos(x) + (3*sin(x) + x*cos(x))*log(x*sin(x)) + -------------------- + --------------------------- + --------------------------------------------| - I*n *(1 + log(x*sin(x))) *(x*cos(x) + sin(x))  + 3*n*(1 + log(x*sin(x)))*(x*cos(x) + sin(x))*|-2*cos(x) + x*sin(x) + (-2*cos(x) + x*sin(x))*log(x*sin(x)) - --------------------||
                         |    |                                                                  2                              2                                 x*sin(x)                  |                                                                                                \                                                                    x*sin(x)      /|
                         \    \                                                                 x *sin(x)                  x*sin (x)                                                        /                                                                                                                                                                                    /
n(xsin⁡(x))inxsin⁡(x)(−in2(xcos⁡(x)+sin⁡(x))3(log⁡(xsin⁡(x))+1)3+3n(xcos⁡(x)+sin⁡(x))(log⁡(xsin⁡(x))+1)(xsin⁡(x)+(xsin⁡(x)−2cos⁡(x))log⁡(xsin⁡(x))−2cos⁡(x)−(xcos⁡(x)+sin⁡(x))2xsin⁡(x))−i(xcos⁡(x)+(xcos⁡(x)+3sin⁡(x))log⁡(xsin⁡(x))+3sin⁡(x)+3(xsin⁡(x)−2cos⁡(x))(xcos⁡(x)+sin⁡(x))xsin⁡(x)+(xcos⁡(x)+sin⁡(x))2cos⁡(x)xsin⁡2(x)+(xcos⁡(x)+sin⁡(x))2x2sin⁡(x)))n \left(x \sin{\left(x \right)}\right)^{i n x \sin{\left(x \right)}} \left(- i n^{2} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \left(\log{\left(x \sin{\left(x \right)} \right)} + 1\right)^{3} + 3 n \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\log{\left(x \sin{\left(x \right)} \right)} + 1\right) \left(x \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} - 2 \cos{\left(x \right)} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2}}{x \sin{\left(x \right)}}\right) - i \left(x \cos{\left(x \right)} + \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} + 3 \sin{\left(x \right)} + \frac{3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2}}{x^{2} \sin{\left(x \right)}}\right)\right)
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