Sr Examen

Derivada de y=ecos^xsenx

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
     x          
E*cos (sin(x))*x
xecosx(sin(x))x e \cos^{x}{\left(\sin{\left(x \right)} \right)}
(E*cos(sin(x))^x)*x
Solución detallada
  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=ecosx(sin(x))f{\left(x \right)} = e \cos^{x}{\left(\sin{\left(x \right)} \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

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

        Perola derivada

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

      Entonces, como resultado: exx(log(x)+1)e x^{x} \left(\log{\left(x \right)} + 1\right)

    g(x)=xg{\left(x \right)} = x; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Según el principio, aplicamos: xx tenemos 11

    Como resultado de: exxx(log(x)+1)+ecosx(sin(x))e x x^{x} \left(\log{\left(x \right)} + 1\right) + e \cos^{x}{\left(\sin{\left(x \right)} \right)}

  2. Simplificamos:

    e(xx+1(log(x)+1)+cosx(sin(x)))e \left(x^{x + 1} \left(\log{\left(x \right)} + 1\right) + \cos^{x}{\left(\sin{\left(x \right)} \right)}\right)


Respuesta:

e(xx+1(log(x)+1)+cosx(sin(x)))e \left(x^{x + 1} \left(\log{\left(x \right)} + 1\right) + \cos^{x}{\left(\sin{\left(x \right)} \right)}\right)

Gráfica
02468-8-6-4-2-1010-1000010000
Primera derivada [src]
     x                  x         /  x*cos(x)*sin(sin(x))                   \
E*cos (sin(x)) + E*x*cos (sin(x))*|- -------------------- + log(cos(sin(x)))|
                                  \      cos(sin(x))                        /
ex(xsin(sin(x))cos(x)cos(sin(x))+log(cos(sin(x))))cosx(sin(x))+ecosx(sin(x))e x \left(- \frac{x \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(\sin{\left(x \right)} \right)} \right)}\right) \cos^{x}{\left(\sin{\left(x \right)} \right)} + e \cos^{x}{\left(\sin{\left(x \right)} \right)}
Segunda derivada [src]
                /                        /                                            2                                           2       2                               \                         \
      x         |                        |  /                    x*cos(x)*sin(sin(x))\         2      2*cos(x)*sin(sin(x))   x*cos (x)*sin (sin(x))   x*sin(x)*sin(sin(x))|   2*x*cos(x)*sin(sin(x))|
-E*cos (sin(x))*|-2*log(cos(sin(x))) + x*|- |-log(cos(sin(x))) + --------------------|  + x*cos (x) + -------------------- + ---------------------- - --------------------| + ----------------------|
                |                        |  \                        cos(sin(x))     /                    cos(sin(x))                2                    cos(sin(x))     |        cos(sin(x))      |
                \                        \                                                                                        cos (sin(x))                            /                         /
e(x(xsin(x)sin(sin(x))cos(sin(x))+xsin2(sin(x))cos2(x)cos2(sin(x))+xcos2(x)(xsin(sin(x))cos(x)cos(sin(x))log(cos(sin(x))))2+2sin(sin(x))cos(x)cos(sin(x)))+2xsin(sin(x))cos(x)cos(sin(x))2log(cos(sin(x))))cosx(sin(x))- e \left(x \left(- \frac{x \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} + \frac{x \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\cos^{2}{\left(\sin{\left(x \right)} \right)}} + x \cos^{2}{\left(x \right)} - \left(\frac{x \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - \log{\left(\cos{\left(\sin{\left(x \right)} \right)} \right)}\right)^{2} + \frac{2 \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}}\right) + \frac{2 x \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - 2 \log{\left(\cos{\left(\sin{\left(x \right)} \right)} \right)}\right) \cos^{x}{\left(\sin{\left(x \right)} \right)}
Tercera derivada [src]
               /                                            2     /                                            3                                                            /                                        2       2                               \        2       2                                                                                    3                         3       3                  2                      \                                               2       2                                 \
     x         |  /                    x*cos(x)*sin(sin(x))\      |  /                    x*cos(x)*sin(sin(x))\         2        /                    x*cos(x)*sin(sin(x))\ |     2      2*cos(x)*sin(sin(x))   x*cos (x)*sin (sin(x))   x*sin(x)*sin(sin(x))|   3*cos (x)*sin (sin(x))                       3*sin(x)*sin(sin(x))   x*cos(x)*sin(sin(x))   2*x*cos (x)*sin(sin(x))   2*x*cos (x)*sin (sin(x))   3*x*sin (sin(x))*cos(x)*sin(x)|          2      6*cos(x)*sin(sin(x))   3*x*cos (x)*sin (sin(x))   3*x*sin(x)*sin(sin(x))|
E*cos (sin(x))*|3*|-log(cos(sin(x))) + --------------------|  + x*|- |-log(cos(sin(x))) + --------------------|  - 3*cos (x) + 3*|-log(cos(sin(x))) + --------------------|*|x*cos (x) + -------------------- + ---------------------- - --------------------| - ---------------------- + 3*x*cos(x)*sin(x) + -------------------- + -------------------- - ----------------------- - ------------------------ + ------------------------------| - 3*x*cos (x) - -------------------- - ------------------------ + ----------------------|
               |  \                        cos(sin(x))     /      |  \                        cos(sin(x))     /                  \                        cos(sin(x))     / |                cos(sin(x))                2                    cos(sin(x))     |           2                                        cos(sin(x))            cos(sin(x))              cos(sin(x))                  3                             2                 |                     cos(sin(x))                 2                      cos(sin(x))      |
               \                                                  \                                                                                                         \                                        cos (sin(x))                            /        cos (sin(x))                                                                                                          cos (sin(x))                  cos (sin(x))         /                                              cos (sin(x))                               /
e(x(3xsin(x)sin2(sin(x))cos(x)cos2(sin(x))+3xsin(x)cos(x)2xsin3(sin(x))cos3(x)cos3(sin(x))2xsin(sin(x))cos3(x)cos(sin(x))+xsin(sin(x))cos(x)cos(sin(x))(xsin(sin(x))cos(x)cos(sin(x))log(cos(sin(x))))3+3(xsin(sin(x))cos(x)cos(sin(x))log(cos(sin(x))))(xsin(x)sin(sin(x))cos(sin(x))+xsin2(sin(x))cos2(x)cos2(sin(x))+xcos2(x)+2sin(sin(x))cos(x)cos(sin(x)))+3sin(x)sin(sin(x))cos(sin(x))3sin2(sin(x))cos2(x)cos2(sin(x))3cos2(x))+3xsin(x)sin(sin(x))cos(sin(x))3xsin2(sin(x))cos2(x)cos2(sin(x))3xcos2(x)+3(xsin(sin(x))cos(x)cos(sin(x))log(cos(sin(x))))26sin(sin(x))cos(x)cos(sin(x)))cosx(sin(x))e \left(x \left(\frac{3 x \sin{\left(x \right)} \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos^{2}{\left(\sin{\left(x \right)} \right)}} + 3 x \sin{\left(x \right)} \cos{\left(x \right)} - \frac{2 x \sin^{3}{\left(\sin{\left(x \right)} \right)} \cos^{3}{\left(x \right)}}{\cos^{3}{\left(\sin{\left(x \right)} \right)}} - \frac{2 x \sin{\left(\sin{\left(x \right)} \right)} \cos^{3}{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} + \frac{x \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - \left(\frac{x \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - \log{\left(\cos{\left(\sin{\left(x \right)} \right)} \right)}\right)^{3} + 3 \left(\frac{x \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - \log{\left(\cos{\left(\sin{\left(x \right)} \right)} \right)}\right) \left(- \frac{x \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} + \frac{x \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\cos^{2}{\left(\sin{\left(x \right)} \right)}} + x \cos^{2}{\left(x \right)} + \frac{2 \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}}\right) + \frac{3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - \frac{3 \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\cos^{2}{\left(\sin{\left(x \right)} \right)}} - 3 \cos^{2}{\left(x \right)}\right) + \frac{3 x \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - \frac{3 x \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\cos^{2}{\left(\sin{\left(x \right)} \right)}} - 3 x \cos^{2}{\left(x \right)} + 3 \left(\frac{x \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}} - \log{\left(\cos{\left(\sin{\left(x \right)} \right)} \right)}\right)^{2} - \frac{6 \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\cos{\left(\sin{\left(x \right)} \right)}}\right) \cos^{x}{\left(\sin{\left(x \right)} \right)}
Gráfico
Derivada de y=ecos^xsenx