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

Derivada de y=sin(sin(sinx^2))

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

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Definida a trozos:

Solución

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

  1. Sustituimos u=sin⁡(sin⁡2(x))u = \sin{\left(\sin^{2}{\left(x \right)} \right)}.

  2. La derivada del seno es igual al coseno:

    ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡(sin⁡2(x))\frac{d}{d x} \sin{\left(\sin^{2}{\left(x \right)} \right)}:

    1. Sustituimos u=sin⁡2(x)u = \sin^{2}{\left(x \right)}.

    2. La derivada del seno es igual al coseno:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡2(x)\frac{d}{d x} \sin^{2}{\left(x \right)}:

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

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

      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:

        2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

      Como resultado de la secuencia de reglas:

      2sin⁡(x)cos⁡(x)cos⁡(sin⁡2(x))2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)}

    Como resultado de la secuencia de reglas:

    2sin⁡(x)cos⁡(x)cos⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x)))2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)}

Respuesta:

2sin⁡(x)cos⁡(x)cos⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x)))2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)}

Gráfica
02468-8-6-4-2-10102-2
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
            /   2   \    /   /   2   \\       
2*cos(x)*cos\sin (x)/*cos\sin\sin (x)//*sin(x)
2sin⁡(x)cos⁡(x)cos⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x)))2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)}
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Segunda derivada [src] 
  /   2       /   2   \    /   /   2   \\      2       /   2   \    /   /   2   \\        2       2/   2   \    2       /   /   2   \\        2       2       /   /   2   \\    /   2   \\
2*\cos (x)*cos\sin (x)/*cos\sin\sin (x)// - sin (x)*cos\sin (x)/*cos\sin\sin (x)// - 2*cos (x)*cos \sin (x)/*sin (x)*sin\sin\sin (x)// - 2*cos (x)*sin (x)*cos\sin\sin (x)//*sin\sin (x)//
2(−2sin⁡2(x)sin⁡(sin⁡2(x))cos⁡2(x)cos⁡(sin⁡(sin⁡2(x)))−2sin⁡2(x)sin⁡(sin⁡(sin⁡2(x)))cos⁡2(x)cos⁡2(sin⁡2(x))−sin⁡2(x)cos⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x)))+cos⁡2(x)cos⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x))))2 \left(- 2 \sin^{2}{\left(x \right)} \sin{\left(\sin^{2}{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} - 2 \sin^{2}{\left(x \right)} \sin{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} \cos^{2}{\left(x \right)} \cos^{2}{\left(\sin^{2}{\left(x \right)} \right)} - \sin^{2}{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} + \cos^{2}{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)}\right)
Simplificar
Tercera derivada [src] 
  /       /   2   \    /   /   2   \\        2       2/   2   \    /   /   2   \\        2       /   /   2   \\    /   2   \        2/   2   \    2       /   /   2   \\        2       /   /   2   \\    /   2   \        2       3/   2   \    2       /   /   2   \\        2       2       /   2   \    /   /   2   \\        2       2       /   2   \    /   2   \    /   /   2   \\\              
4*\- 2*cos\sin (x)/*cos\sin\sin (x)// - 3*cos (x)*cos \sin (x)/*sin\sin\sin (x)// - 3*cos (x)*cos\sin\sin (x)//*sin\sin (x)/ + 3*cos \sin (x)/*sin (x)*sin\sin\sin (x)// + 3*sin (x)*cos\sin\sin (x)//*sin\sin (x)/ - 2*cos (x)*cos \sin (x)/*sin (x)*cos\sin\sin (x)// - 2*cos (x)*sin (x)*cos\sin (x)/*cos\sin\sin (x)// + 6*cos (x)*sin (x)*cos\sin (x)/*sin\sin (x)/*sin\sin\sin (x)///*cos(x)*sin(x)
4(6sin⁡2(x)sin⁡(sin⁡2(x))sin⁡(sin⁡(sin⁡2(x)))cos⁡2(x)cos⁡(sin⁡2(x))+3sin⁡2(x)sin⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x)))+3sin⁡2(x)sin⁡(sin⁡(sin⁡2(x)))cos⁡2(sin⁡2(x))−2sin⁡2(x)cos⁡2(x)cos⁡3(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x)))−2sin⁡2(x)cos⁡2(x)cos⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x)))−3sin⁡(sin⁡2(x))cos⁡2(x)cos⁡(sin⁡(sin⁡2(x)))−3sin⁡(sin⁡(sin⁡2(x)))cos⁡2(x)cos⁡2(sin⁡2(x))−2cos⁡(sin⁡2(x))cos⁡(sin⁡(sin⁡2(x))))sin⁡(x)cos⁡(x)4 \left(6 \sin^{2}{\left(x \right)} \sin{\left(\sin^{2}{\left(x \right)} \right)} \sin{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)} + 3 \sin^{2}{\left(x \right)} \sin{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} + 3 \sin^{2}{\left(x \right)} \sin{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\sin^{2}{\left(x \right)} \right)} - 2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \cos^{3}{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} - 2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} - 3 \sin{\left(\sin^{2}{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} - 3 \sin{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)} \cos^{2}{\left(x \right)} \cos^{2}{\left(\sin^{2}{\left(x \right)} \right)} - 2 \cos{\left(\sin^{2}{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin^{2}{\left(x \right)} \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}
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Gráfico
Derivada de y=sin(sin(sinx^2))
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