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y=cos^2(sin^4)
Derivada de la función/ y=cos/ cos^2/ sin^4/ y=cos^2(sin^4)

Derivada de y=cos^2(sin^4)

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src] 
   2/   4   \
cos \sin (x)/
cos⁡2(sin⁡4(x))\cos^{2}{\left(\sin^{4}{\left(x \right)} \right)} 
cos(sin(x)^4)^2
Solución detallada

  1. Sustituimos u=cos⁡(sin⁡4(x))u = \cos{\left(\sin^{4}{\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⁡(sin⁡4(x))\frac{d}{d x} \cos{\left(\sin^{4}{\left(x \right)} \right)}:

    1. Sustituimos u=sin⁡4(x)u = \sin^{4}{\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 ddxsin⁡4(x)\frac{d}{d x} \sin^{4}{\left(x \right)}:

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

      2. Según el principio, aplicamos: u4u^{4} tenemos 4u34 u^{3}

      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:

        4sin⁡3(x)cos⁡(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}

      Como resultado de la secuencia de reglas:

      −4sin⁡3(x)sin⁡(sin⁡4(x))cos⁡(x)- 4 \sin^{3}{\left(x \right)} \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos{\left(x \right)}

    Como resultado de la secuencia de reglas:

    −8sin⁡3(x)sin⁡(sin⁡4(x))cos⁡(x)cos⁡(sin⁡4(x))- 8 \sin^{3}{\left(x \right)} \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)}

Respuesta:

−8sin⁡3(x)sin⁡(sin⁡4(x))cos⁡(x)cos⁡(sin⁡4(x))- 8 \sin^{3}{\left(x \right)} \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)}

Gráfica
02468-8-6-4-2-10102.5-2.5
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Primera derivada [src] 
      3              /   4   \    /   4   \
-8*sin (x)*cos(x)*cos\sin (x)/*sin\sin (x)/
−8sin⁡3(x)sin⁡(sin⁡4(x))cos⁡(x)cos⁡(sin⁡4(x))- 8 \sin^{3}{\left(x \right)} \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)}
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Segunda derivada [src] 
     2    /   2       /   4   \    /   4   \        2       2/   4   \    4           2       /   4   \    /   4   \        2       4       2/   4   \\
8*sin (x)*\sin (x)*cos\sin (x)/*sin\sin (x)/ - 4*cos (x)*cos \sin (x)/*sin (x) - 3*cos (x)*cos\sin (x)/*sin\sin (x)/ + 4*cos (x)*sin (x)*sin \sin (x)//
8(4sin⁡4(x)sin⁡2(sin⁡4(x))cos⁡2(x)−4sin⁡4(x)cos⁡2(x)cos⁡2(sin⁡4(x))+sin⁡2(x)sin⁡(sin⁡4(x))cos⁡(sin⁡4(x))−3sin⁡(sin⁡4(x))cos⁡2(x)cos⁡(sin⁡4(x)))sin⁡2(x)8 \left(4 \sin^{4}{\left(x \right)} \sin^{2}{\left(\sin^{4}{\left(x \right)} \right)} \cos^{2}{\left(x \right)} - 4 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} \cos^{2}{\left(\sin^{4}{\left(x \right)} \right)} + \sin^{2}{\left(x \right)} \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)} - 3 \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)}\right) \sin^{2}{\left(x \right)}
Simplificar
Tercera derivada [src] 
   /       6       2/   4   \        2/   4   \    6            2       2/   4   \    4           2       /   4   \    /   4   \        2       /   4   \    /   4   \         2       4       2/   4   \         2       8       /   4   \    /   4   \\              
16*\- 6*sin (x)*sin \sin (x)/ + 6*cos \sin (x)/*sin (x) - 18*cos (x)*cos \sin (x)/*sin (x) - 3*cos (x)*cos\sin (x)/*sin\sin (x)/ + 5*sin (x)*cos\sin (x)/*sin\sin (x)/ + 18*cos (x)*sin (x)*sin \sin (x)/ + 32*cos (x)*sin (x)*cos\sin (x)/*sin\sin (x)//*cos(x)*sin(x)
16(32sin⁡8(x)sin⁡(sin⁡4(x))cos⁡2(x)cos⁡(sin⁡4(x))−6sin⁡6(x)sin⁡2(sin⁡4(x))+6sin⁡6(x)cos⁡2(sin⁡4(x))+18sin⁡4(x)sin⁡2(sin⁡4(x))cos⁡2(x)−18sin⁡4(x)cos⁡2(x)cos⁡2(sin⁡4(x))+5sin⁡2(x)sin⁡(sin⁡4(x))cos⁡(sin⁡4(x))−3sin⁡(sin⁡4(x))cos⁡2(x)cos⁡(sin⁡4(x)))sin⁡(x)cos⁡(x)16 \left(32 \sin^{8}{\left(x \right)} \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)} - 6 \sin^{6}{\left(x \right)} \sin^{2}{\left(\sin^{4}{\left(x \right)} \right)} + 6 \sin^{6}{\left(x \right)} \cos^{2}{\left(\sin^{4}{\left(x \right)} \right)} + 18 \sin^{4}{\left(x \right)} \sin^{2}{\left(\sin^{4}{\left(x \right)} \right)} \cos^{2}{\left(x \right)} - 18 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} \cos^{2}{\left(\sin^{4}{\left(x \right)} \right)} + 5 \sin^{2}{\left(x \right)} \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)} - 3 \sin{\left(\sin^{4}{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin^{4}{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}
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Gráfico
Derivada de y=cos^2(sin^4)
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