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y=cos^5(tan(sin(x^5)))
Derivada de la función/ (x^5)/ y=cos/ y=cos^5(tan(sin(x^5)))

Derivada de y=cos^5(tan(sin(x^5)))

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

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Solución

Ha introducido [src] 
   5/   /   / 5\\\
cos \tan\sin\x ///
cos⁡5(tan⁡(sin⁡(x5)))\cos^{5}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} 
cos(tan(sin(x^5)))^5
Solución detallada

  1. Sustituimos u=cos⁡(tan⁡(sin⁡(x5)))u = \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}.

  2. Según el principio, aplicamos: u5u^{5} tenemos 5u45 u^{4}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxcos⁡(tan⁡(sin⁡(x5)))\frac{d}{d x} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}:

    1. Sustituimos u=tan⁡(sin⁡(x5))u = \tan{\left(\sin{\left(x^{5} \right)} \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⁡(sin⁡(x5))\frac{d}{d x} \tan{\left(\sin{\left(x^{5} \right)} \right)}:

      1. Reescribimos las funciones para diferenciar:

        tan⁡(sin⁡(x5))=sin⁡(sin⁡(x5))cos⁡(sin⁡(x5))\tan{\left(\sin{\left(x^{5} \right)} \right)} = \frac{\sin{\left(\sin{\left(x^{5} \right)} \right)}}{\cos{\left(\sin{\left(x^{5} \right)} \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⁡(sin⁡(x5))f{\left(x \right)} = \sin{\left(\sin{\left(x^{5} \right)} \right)} y g(x)=cos⁡(sin⁡(x5))g{\left(x \right)} = \cos{\left(\sin{\left(x^{5} \right)} \right)}.

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

        1. Sustituimos u=sin⁡(x5)u = \sin{\left(x^{5} \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⁡(x5)\frac{d}{d x} \sin{\left(x^{5} \right)}:

          1. Sustituimos u=x5u = x^{5}.

          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 ddxx5\frac{d}{d x} x^{5}:

            1. Según el principio, aplicamos: x5x^{5} tenemos 5x45 x^{4}

            Como resultado de la secuencia de reglas:

            5x4cos⁡(x5)5 x^{4} \cos{\left(x^{5} \right)}

          Como resultado de la secuencia de reglas:

          5x4cos⁡(x5)cos⁡(sin⁡(x5))5 x^{4} \cos{\left(x^{5} \right)} \cos{\left(\sin{\left(x^{5} \right)} \right)}

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

        1. Sustituimos u=sin⁡(x5)u = \sin{\left(x^{5} \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⁡(x5)\frac{d}{d x} \sin{\left(x^{5} \right)}:

          1. Sustituimos u=x5u = x^{5}.

          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 ddxx5\frac{d}{d x} x^{5}:

            1. Según el principio, aplicamos: x5x^{5} tenemos 5x45 x^{4}

            Como resultado de la secuencia de reglas:

            5x4cos⁡(x5)5 x^{4} \cos{\left(x^{5} \right)}

          Como resultado de la secuencia de reglas:

          −5x4sin⁡(sin⁡(x5))cos⁡(x5)- 5 x^{4} \sin{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)}

        Ahora aplicamos la regla de la derivada de una divesión:

        5x4sin⁡2(sin⁡(x5))cos⁡(x5)+5x4cos⁡(x5)cos⁡2(sin⁡(x5))cos⁡2(sin⁡(x5))\frac{5 x^{4} \sin^{2}{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)} + 5 x^{4} \cos{\left(x^{5} \right)} \cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}

      Como resultado de la secuencia de reglas:

      −(5x4sin⁡2(sin⁡(x5))cos⁡(x5)+5x4cos⁡(x5)cos⁡2(sin⁡(x5)))sin⁡(tan⁡(sin⁡(x5)))cos⁡2(sin⁡(x5))- \frac{\left(5 x^{4} \sin^{2}{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)} + 5 x^{4} \cos{\left(x^{5} \right)} \cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}

    Como resultado de la secuencia de reglas:

    −5(5x4sin⁡2(sin⁡(x5))cos⁡(x5)+5x4cos⁡(x5)cos⁡2(sin⁡(x5)))sin⁡(tan⁡(sin⁡(x5)))cos⁡4(tan⁡(sin⁡(x5)))cos⁡2(sin⁡(x5))- \frac{5 \left(5 x^{4} \sin^{2}{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)} + 5 x^{4} \cos{\left(x^{5} \right)} \cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}

  4. Simplificamos:

    −25x4sin⁡(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡4(tan⁡(sin⁡(x5)))cos⁡2(sin⁡(x5))- \frac{25 x^{4} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}

Respuesta:

−25x4sin⁡(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡4(tan⁡(sin⁡(x5)))cos⁡2(sin⁡(x5))- \frac{25 x^{4} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}

Gráfica
02468-8-6-4-2-1010-100000100000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
     4    4/   /   / 5\\\ /       2/   / 5\\\    / 5\    /   /   / 5\\\
-25*x *cos \tan\sin\x ///*\1 + tan \sin\x ///*cos\x /*sin\tan\sin\x ///
−25x4(tan⁡2(sin⁡(x5))+1)sin⁡(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡4(tan⁡(sin⁡(x5)))- 25 x^{4} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}
Simplificar
Segunda derivada [src] 
    3    3/   /   / 5\\\ /       2/   / 5\\\ /       / 5\    /   /   / 5\\\    /   /   / 5\\\      5    2/ 5\    2/   /   / 5\\\ /       2/   / 5\\\      5    /   /   / 5\\\    / 5\    /   /   / 5\\\       5    2/ 5\    2/   /   / 5\\\ /       2/   / 5\\\       5    2/ 5\    /   /   / 5\\\    /   /   / 5\\\    /   / 5\\\
25*x *cos \tan\sin\x ///*\1 + tan \sin\x ///*\- 4*cos\x /*cos\tan\sin\x ///*sin\tan\sin\x /// - 5*x *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x /// + 5*x *cos\tan\sin\x ///*sin\x /*sin\tan\sin\x /// + 20*x *cos \x /*sin \tan\sin\x ///*\1 + tan \sin\x /// - 10*x *cos \x /*cos\tan\sin\x ///*sin\tan\sin\x ///*tan\sin\x ///
25x3(tan⁡2(sin⁡(x5))+1)(20x5(tan⁡2(sin⁡(x5))+1)sin⁡2(tan⁡(sin⁡(x5)))cos⁡2(x5)−5x5(tan⁡2(sin⁡(x5))+1)cos⁡2(x5)cos⁡2(tan⁡(sin⁡(x5)))+5x5sin⁡(x5)sin⁡(tan⁡(sin⁡(x5)))cos⁡(tan⁡(sin⁡(x5)))−10x5sin⁡(tan⁡(sin⁡(x5)))cos⁡2(x5)cos⁡(tan⁡(sin⁡(x5)))tan⁡(sin⁡(x5))−4sin⁡(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡(tan⁡(sin⁡(x5))))cos⁡3(tan⁡(sin⁡(x5)))25 x^{3} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \left(20 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} - 5 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \cos^{2}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 5 x^{5} \sin{\left(x^{5} \right)} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 10 x^{5} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 4 \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}\right) \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}
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Tercera derivada [src] 
                                             /                                                                               2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2                                                                                                                                                                                                                                                                                                      \
    2    2/   /   / 5\\\ /       2/   / 5\\\ |        2/   /   / 5\\\    / 5\    /   /   / 5\\\        10 /       2/   / 5\\\     3/ 5\    3/   /   / 5\\\       5    2/ 5\    3/   /   / 5\\\ /       2/   / 5\\\       10    2/   /   / 5\\\    / 5\    /   /   / 5\\\       5    2/   /   / 5\\\    / 5\    /   /   / 5\\\        10    3/ 5\    3/   /   / 5\\\ /       2/   / 5\\\    /   / 5\\        5    2/ 5\    2/   /   / 5\\\    /   /   / 5\\\    /   / 5\\        10    3/ 5\    2/   /   / 5\\\    2/   / 5\\    /   /   / 5\\\       10    3/ 5\    2/   /   / 5\\\ /       2/   / 5\\\    /   /   / 5\\\       10    3/   /   / 5\\\ /       2/   / 5\\\    / 5\    / 5\        5    2/ 5\    2/   /   / 5\\\ /       2/   / 5\\\    /   /   / 5\\\        10 /       2/   / 5\\\     3/ 5\    2/   /   / 5\\\    /   /   / 5\\\        10    2/   /   / 5\\\ /       2/   / 5\\\    / 5\    /   /   / 5\\\    / 5\        10    2/   /   / 5\\\    / 5\    / 5\    /   /   / 5\\\    /   / 5\\        10    3/ 5\    2/   /   / 5\\\ /       2/   / 5\\\    /   /   / 5\\\    /   / 5\\|
25*x *cos \tan\sin\x ///*\1 + tan \sin\x ///*\- 12*cos \tan\sin\x ///*cos\x /*sin\tan\sin\x /// - 300*x  *\1 + tan \sin\x /// *cos \x /*sin \tan\sin\x /// - 60*x *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x /// + 25*x  *cos \tan\sin\x ///*cos\x /*sin\tan\sin\x /// + 60*x *cos \tan\sin\x ///*sin\x /*sin\tan\sin\x /// - 150*x  *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x ///*tan\sin\x // - 120*x *cos \x /*cos \tan\sin\x ///*sin\tan\sin\x ///*tan\sin\x // - 100*x  *cos \x /*cos \tan\sin\x ///*tan \sin\x //*sin\tan\sin\x /// - 50*x  *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x ///*sin\tan\sin\x /// + 75*x  *cos \tan\sin\x ///*\1 + tan \sin\x ///*cos\x /*sin\x / + 240*x *cos \x /*sin \tan\sin\x ///*\1 + tan \sin\x ///*cos\tan\sin\x /// + 325*x  *\1 + tan \sin\x /// *cos \x /*cos \tan\sin\x ///*sin\tan\sin\x /// - 300*x  *sin \tan\sin\x ///*\1 + tan \sin\x ///*cos\x /*cos\tan\sin\x ///*sin\x / + 150*x  *cos \tan\sin\x ///*cos\x /*sin\x /*sin\tan\sin\x ///*tan\sin\x // + 600*x  *cos \x /*sin \tan\sin\x ///*\1 + tan \sin\x ///*cos\tan\sin\x ///*tan\sin\x ///
25x2(tan⁡2(sin⁡(x5))+1)(−300x10(tan⁡2(sin⁡(x5))+1)2sin⁡3(tan⁡(sin⁡(x5)))cos⁡3(x5)+325x10(tan⁡2(sin⁡(x5))+1)2sin⁡(tan⁡(sin⁡(x5)))cos⁡3(x5)cos⁡2(tan⁡(sin⁡(x5)))−300x10(tan⁡2(sin⁡(x5))+1)sin⁡(x5)sin⁡2(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡(tan⁡(sin⁡(x5)))+75x10(tan⁡2(sin⁡(x5))+1)sin⁡(x5)cos⁡(x5)cos⁡3(tan⁡(sin⁡(x5)))+600x10(tan⁡2(sin⁡(x5))+1)sin⁡2(tan⁡(sin⁡(x5)))cos⁡3(x5)cos⁡(tan⁡(sin⁡(x5)))tan⁡(sin⁡(x5))−50x10(tan⁡2(sin⁡(x5))+1)sin⁡(tan⁡(sin⁡(x5)))cos⁡3(x5)cos⁡2(tan⁡(sin⁡(x5)))−150x10(tan⁡2(sin⁡(x5))+1)cos⁡3(x5)cos⁡3(tan⁡(sin⁡(x5)))tan⁡(sin⁡(x5))+150x10sin⁡(x5)sin⁡(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡2(tan⁡(sin⁡(x5)))tan⁡(sin⁡(x5))−100x10sin⁡(tan⁡(sin⁡(x5)))cos⁡3(x5)cos⁡2(tan⁡(sin⁡(x5)))tan⁡2(sin⁡(x5))+25x10sin⁡(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡2(tan⁡(sin⁡(x5)))+240x5(tan⁡2(sin⁡(x5))+1)sin⁡2(tan⁡(sin⁡(x5)))cos⁡2(x5)cos⁡(tan⁡(sin⁡(x5)))−60x5(tan⁡2(sin⁡(x5))+1)cos⁡2(x5)cos⁡3(tan⁡(sin⁡(x5)))+60x5sin⁡(x5)sin⁡(tan⁡(sin⁡(x5)))cos⁡2(tan⁡(sin⁡(x5)))−120x5sin⁡(tan⁡(sin⁡(x5)))cos⁡2(x5)cos⁡2(tan⁡(sin⁡(x5)))tan⁡(sin⁡(x5))−12sin⁡(tan⁡(sin⁡(x5)))cos⁡(x5)cos⁡2(tan⁡(sin⁡(x5))))cos⁡2(tan⁡(sin⁡(x5)))25 x^{2} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \left(- 300 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right)^{2} \sin^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} + 325 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right)^{2} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 300 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(x^{5} \right)} \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 75 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(x^{5} \right)} \cos{\left(x^{5} \right)} \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 600 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 50 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 150 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \cos^{3}{\left(x^{5} \right)} \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} + 150 x^{10} \sin{\left(x^{5} \right)} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 100 x^{10} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 25 x^{10} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 240 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 60 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \cos^{2}{\left(x^{5} \right)} \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 60 x^{5} \sin{\left(x^{5} \right)} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 120 x^{5} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 12 \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}\right) \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}
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Gráfico
Derivada de y=cos^5(tan(sin(x^5)))
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