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sin(cos(tan(x)^(3))^(2))
Derivada de la función/ tan(x)/ sin(cos(tan(x)^(3))^(2))

Derivada de sin(cos(tan(x)^(3))^(2))

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

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Solución

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

  1. Sustituimos u=cos⁡2(tan⁡3(x))u = \cos^{2}{\left(\tan^{3}{\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 ddxcos⁡2(tan⁡3(x))\frac{d}{d x} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)}:

    1. Sustituimos u=cos⁡(tan⁡3(x))u = \cos{\left(\tan^{3}{\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⁡(tan⁡3(x))\frac{d}{d x} \cos{\left(\tan^{3}{\left(x \right)} \right)}:

      1. Sustituimos u=tan⁡3(x)u = \tan^{3}{\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 ddxtan⁡3(x)\frac{d}{d x} \tan^{3}{\left(x \right)}:

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

        2. Según el principio, aplicamos: u3u^{3} tenemos 3u23 u^{2}

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

          1. Reescribimos las funciones para diferenciar:

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

          2. Se aplica la regla de la derivada parcial:

            ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

            f(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} y g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

            Para calcular ddxf(x)\frac{d}{d x} f{\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)}

            Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

              ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

            Ahora aplicamos la regla de la derivada de una divesión:

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

          Como resultado de la secuencia de reglas:

          3(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        −3(sin⁡2(x)+cos⁡2(x))sin⁡(tan⁡3(x))tan⁡2(x)cos⁡2(x)- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      Como resultado de la secuencia de reglas:

      −6(sin⁡2(x)+cos⁡2(x))sin⁡(tan⁡3(x))cos⁡(tan⁡3(x))tan⁡2(x)cos⁡2(x)- \frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    Como resultado de la secuencia de reglas:

    −6(sin⁡2(x)+cos⁡2(x))sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡2(x)cos⁡2(x)- \frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  4. Simplificamos:

    −6sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡2(x)cos⁡2(x)- \frac{6 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

Respuesta:

−6sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡2(x)cos⁡2(x)- \frac{6 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-50000005000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
      2    /         2   \    /   2/   3   \\    /   3   \    /   3   \
-2*tan (x)*\3 + 3*tan (x)/*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/
−2(3tan⁡2(x)+3)sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡2(x)- 2 \left(3 \tan^{2}{\left(x \right)} + 3\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}
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Segunda derivada [src] 
  /       2   \ /       2/   3   \    3    /       2   \    /   2/   3   \\        2       /   2/   3   \\    /   3   \    /   3   \     /       2   \    /   2/   3   \\    /   3   \    /   3   \        2/   3   \    3    /       2   \    /   2/   3   \\        2/   3   \    2/   3   \    3    /       2   \    /   2/   3   \\\       
6*\1 + tan (x)/*\- 3*cos \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 2*tan (x)*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ - 2*\1 + tan (x)/*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 3*sin \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 6*cos \tan (x)/*sin \tan (x)/*tan (x)*\1 + tan (x)/*sin\cos \tan (x)///*tan(x)
6(tan⁡2(x)+1)(−6(tan⁡2(x)+1)sin⁡(cos⁡2(tan⁡3(x)))sin⁡2(tan⁡3(x))cos⁡2(tan⁡3(x))tan⁡3(x)+3(tan⁡2(x)+1)sin⁡2(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))tan⁡3(x)−2(tan⁡2(x)+1)sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))−3(tan⁡2(x)+1)cos⁡(cos⁡2(tan⁡3(x)))cos⁡2(tan⁡3(x))tan⁡3(x)−2sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡2(x))tan⁡(x)6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \tan^{3}{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 2 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}\right) \tan{\left(x \right)}
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Tercera derivada [src] 
                 /               2                                                               2                                                                                                                                                                               2                                                                                                                       2                                                                         2                                                                                                                                                                                                                           2                                                                          2                                                                        2                                                      \
   /       2   \ |  /       2   \     /   2/   3   \\    /   3   \    /   3   \     /       2   \     2/   3   \    3       /   2/   3   \\        2/   3   \    5    /       2   \    /   2/   3   \\        4       /   2/   3   \\    /   3   \    /   3   \     /       2   \     2/   3   \    3       /   2/   3   \\        2/   3   \    5    /       2   \    /   2/   3   \\      /       2   \     3/   3   \    6       /   2/   3   \\    /   3   \      /       2   \     2/   3   \    2/   3   \    3       /   2/   3   \\         2/   3   \    2/   3   \    5    /       2   \    /   2/   3   \\        2    /       2   \    /   2/   3   \\    /   3   \    /   3   \      /       2   \     3/   3   \    3/   3   \    6       /   2/   3   \\      /       2   \     6       /   2/   3   \\    /   3   \    /   3   \      /       2   \     3/   3   \    6       /   3   \    /   2/   3   \\|
12*\1 + tan (x)/*\- \1 + tan (x)/ *cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ - 9*\1 + tan (x)/ *cos \tan (x)/*tan (x)*cos\cos \tan (x)// - 9*cos \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 2*tan (x)*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 9*\1 + tan (x)/ *sin \tan (x)/*tan (x)*cos\cos \tan (x)// + 9*sin \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 27*\1 + tan (x)/ *cos \tan (x)/*tan (x)*sin\cos \tan (x)//*sin\tan (x)/ - 18*\1 + tan (x)/ *cos \tan (x)/*sin \tan (x)/*tan (x)*sin\cos \tan (x)// - 18*cos \tan (x)/*sin \tan (x)/*tan (x)*\1 + tan (x)/*sin\cos \tan (x)// - 7*tan (x)*\1 + tan (x)/*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 18*\1 + tan (x)/ *cos \tan (x)/*sin \tan (x)/*tan (x)*cos\cos \tan (x)// + 18*\1 + tan (x)/ *tan (x)*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 27*\1 + tan (x)/ *sin \tan (x)/*tan (x)*cos\tan (x)/*sin\cos \tan (x)///
12(tan⁡2(x)+1)(27(tan⁡2(x)+1)2sin⁡(cos⁡2(tan⁡3(x)))sin⁡3(tan⁡3(x))cos⁡(tan⁡3(x))tan⁡6(x)−18(tan⁡2(x)+1)2sin⁡(cos⁡2(tan⁡3(x)))sin⁡2(tan⁡3(x))cos⁡2(tan⁡3(x))tan⁡3(x)−27(tan⁡2(x)+1)2sin⁡(cos⁡2(tan⁡3(x)))sin⁡(tan⁡3(x))cos⁡3(tan⁡3(x))tan⁡6(x)+18(tan⁡2(x)+1)2sin⁡3(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡3(tan⁡3(x))tan⁡6(x)+9(tan⁡2(x)+1)2sin⁡2(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))tan⁡3(x)+18(tan⁡2(x)+1)2sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡6(x)−(tan⁡2(x)+1)2sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))−9(tan⁡2(x)+1)2cos⁡(cos⁡2(tan⁡3(x)))cos⁡2(tan⁡3(x))tan⁡3(x)−18(tan⁡2(x)+1)sin⁡(cos⁡2(tan⁡3(x)))sin⁡2(tan⁡3(x))cos⁡2(tan⁡3(x))tan⁡5(x)+9(tan⁡2(x)+1)sin⁡2(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))tan⁡5(x)−7(tan⁡2(x)+1)sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡2(x)−9(tan⁡2(x)+1)cos⁡(cos⁡2(tan⁡3(x)))cos⁡2(tan⁡3(x))tan⁡5(x)−2sin⁡(tan⁡3(x))cos⁡(cos⁡2(tan⁡3(x)))cos⁡(tan⁡3(x))tan⁡4(x))12 \left(\tan^{2}{\left(x \right)} + 1\right) \left(27 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} - 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 27 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} + 9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \tan^{3}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 18 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} + 9 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \tan^{5}{\left(x \right)} - 7 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} - 2 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{4}{\left(x \right)}\right)
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Gráfico
Derivada de sin(cos(tan(x)^(3))^(2))
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