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y=cos^2*(3^sinx)
Derivada de la función/ cos^2/ y=cos/ 3^sinx/ y=cos^2*(3^sinx)

Derivada de y=cos^2*(3^sinx)

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

Ha introducido [src] 
   2/ sin(x)\
cos \3      /
cos⁡2(3sin⁡(x))\cos^{2}{\left(3^{\sin{\left(x \right)}} \right)} 
cos(3^sin(x))^2
Solución detallada

  1. Sustituimos u=cos⁡(3sin⁡(x))u = \cos{\left(3^{\sin{\left(x \right)}} \right)}.

  2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxcos⁡(3sin⁡(x))\frac{d}{d x} \cos{\left(3^{\sin{\left(x \right)}} \right)}:

    1. Sustituimos u=3sin⁡(x)u = 3^{\sin{\left(x \right)}}.

    2. La derivada del coseno es igual a menos el seno:

      dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddx3sin⁡(x)\frac{d}{d x} 3^{\sin{\left(x \right)}}:

      1. Sustituimos u=sin⁡(x)u = \sin{\left(x \right)}.

      2. ddu3u=3ulog⁡(3)\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

        1. La derivada del seno es igual al coseno:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        Como resultado de la secuencia de reglas:

        3sin⁡(x)log⁡(3)cos⁡(x)3^{\sin{\left(x \right)}} \log{\left(3 \right)} \cos{\left(x \right)}

      Como resultado de la secuencia de reglas:

      −3sin⁡(x)log⁡(3)sin⁡(3sin⁡(x))cos⁡(x)- 3^{\sin{\left(x \right)}} \log{\left(3 \right)} \sin{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)}

    Como resultado de la secuencia de reglas:

    −2⋅3sin⁡(x)log⁡(3)sin⁡(3sin⁡(x))cos⁡(3sin⁡(x))cos⁡(x)- 2 \cdot 3^{\sin{\left(x \right)}} \log{\left(3 \right)} \sin{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)}

  4. Simplificamos:

    −3sin⁡(x)(sin⁡(2⋅3sin⁡(x)−x)+sin⁡(2⋅3sin⁡(x)+x))log⁡(3)2- \frac{3^{\sin{\left(x \right)}} \left(\sin{\left(2 \cdot 3^{\sin{\left(x \right)}} - x \right)} + \sin{\left(2 \cdot 3^{\sin{\left(x \right)}} + x \right)}\right) \log{\left(3 \right)}}{2}

Respuesta:

−3sin⁡(x)(sin⁡(2⋅3sin⁡(x)−x)+sin⁡(2⋅3sin⁡(x)+x))log⁡(3)2- \frac{3^{\sin{\left(x \right)}} \left(\sin{\left(2 \cdot 3^{\sin{\left(x \right)}} - x \right)} + \sin{\left(2 \cdot 3^{\sin{\left(x \right)}} + x \right)}\right) \log{\left(3 \right)}}{2}

Gráfica
02468-8-6-4-2-10105-5
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
    sin(x)           / sin(x)\           / sin(x)\
-2*3      *cos(x)*cos\3      /*log(3)*sin\3      /
−2⋅3sin⁡(x)log⁡(3)sin⁡(3sin⁡(x))cos⁡(3sin⁡(x))cos⁡(x)- 2 \cdot 3^{\sin{\left(x \right)}} \log{\left(3 \right)} \sin{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)}
Simplificar
Segunda derivada [src] 
   sin(x) /   / sin(x)\           / sin(x)\    sin(x)    2       2/ sin(x)\           sin(x)    2       2/ sin(x)\             2       / sin(x)\           / sin(x)\\       
2*3      *\cos\3      /*sin(x)*sin\3      / + 3      *cos (x)*sin \3      /*log(3) - 3      *cos (x)*cos \3      /*log(3) - cos (x)*cos\3      /*log(3)*sin\3      //*log(3)
2⋅3sin⁡(x)(3sin⁡(x)log⁡(3)sin⁡2(3sin⁡(x))cos⁡2(x)−3sin⁡(x)log⁡(3)cos⁡2(3sin⁡(x))cos⁡2(x)+sin⁡(3sin⁡(x))sin⁡(x)cos⁡(3sin⁡(x))−log⁡(3)sin⁡(3sin⁡(x))cos⁡(3sin⁡(x))cos⁡2(x))log⁡(3)2 \cdot 3^{\sin{\left(x \right)}} \left(3^{\sin{\left(x \right)}} \log{\left(3 \right)} \sin^{2}{\left(3^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} - 3^{\sin{\left(x \right)}} \log{\left(3 \right)} \cos^{2}{\left(3^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + \sin{\left(3^{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)} - \log{\left(3 \right)} \sin{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)}\right) \log{\left(3 \right)}
Simplificar
Tercera derivada [src] 
   sin(x) /   / sin(x)\    / sin(x)\      2       2       / sin(x)\    / sin(x)\      sin(x)    2       2/ sin(x)\    2         sin(x)    2/ sin(x)\                    sin(x)    2       2       2/ sin(x)\      sin(x)    2/ sin(x)\                      / sin(x)\                  / sin(x)\      2*sin(x)    2       2       / sin(x)\    / sin(x)\\              
2*3      *\cos\3      /*sin\3      / - cos (x)*log (3)*cos\3      /*sin\3      / - 3*3      *cos (x)*cos \3      /*log (3) - 3*3      *sin \3      /*log(3)*sin(x) + 3*3      *cos (x)*log (3)*sin \3      / + 3*3      *cos \3      /*log(3)*sin(x) + 3*cos\3      /*log(3)*sin(x)*sin\3      / + 4*3        *cos (x)*log (3)*cos\3      /*sin\3      //*cos(x)*log(3)
2⋅3sin⁡(x)(4⋅32sin⁡(x)log⁡(3)2sin⁡(3sin⁡(x))cos⁡(3sin⁡(x))cos⁡2(x)−3⋅3sin⁡(x)log⁡(3)sin⁡2(3sin⁡(x))sin⁡(x)+3⋅3sin⁡(x)log⁡(3)2sin⁡2(3sin⁡(x))cos⁡2(x)+3⋅3sin⁡(x)log⁡(3)sin⁡(x)cos⁡2(3sin⁡(x))−3⋅3sin⁡(x)log⁡(3)2cos⁡2(3sin⁡(x))cos⁡2(x)+3log⁡(3)sin⁡(3sin⁡(x))sin⁡(x)cos⁡(3sin⁡(x))−log⁡(3)2sin⁡(3sin⁡(x))cos⁡(3sin⁡(x))cos⁡2(x)+sin⁡(3sin⁡(x))cos⁡(3sin⁡(x)))log⁡(3)cos⁡(x)2 \cdot 3^{\sin{\left(x \right)}} \left(4 \cdot 3^{2 \sin{\left(x \right)}} \log{\left(3 \right)}^{2} \sin{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} - 3 \cdot 3^{\sin{\left(x \right)}} \log{\left(3 \right)} \sin^{2}{\left(3^{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} + 3 \cdot 3^{\sin{\left(x \right)}} \log{\left(3 \right)}^{2} \sin^{2}{\left(3^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + 3 \cdot 3^{\sin{\left(x \right)}} \log{\left(3 \right)} \sin{\left(x \right)} \cos^{2}{\left(3^{\sin{\left(x \right)}} \right)} - 3 \cdot 3^{\sin{\left(x \right)}} \log{\left(3 \right)}^{2} \cos^{2}{\left(3^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + 3 \log{\left(3 \right)} \sin{\left(3^{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)} - \log{\left(3 \right)}^{2} \sin{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + \sin{\left(3^{\sin{\left(x \right)}} \right)} \cos{\left(3^{\sin{\left(x \right)}} \right)}\right) \log{\left(3 \right)} \cos{\left(x \right)}
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Gráfico
Derivada de y=cos^2*(3^sinx)
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