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y=(sinx+tgx)^16
Derivada de la función/ x+tgx/ y=(sinx+tgx)^16

Derivada de y=(sinx+tgx)^16

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

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Solución

Ha introducido [src] 
                 16
(sin(x) + tan(x))
(sin⁡(x)+tan⁡(x))16\left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{16} 
(sin(x) + tan(x))^16
Solución detallada

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

  2. Según el principio, aplicamos: u16u^{16} tenemos 16u1516 u^{15}

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

    1. diferenciamos sin⁡(x)+tan⁡(x)\sin{\left(x \right)} + \tan{\left(x \right)} miembro por miembro:

      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)}

      2. Reescribimos las funciones para diferenciar:

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

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

    Como resultado de la secuencia de reglas:

    16(sin⁡2(x)+cos⁡2(x)cos⁡2(x)+cos⁡(x))(sin⁡(x)+tan⁡(x))1516 \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{15}

  4. Simplificamos:

    16(sin⁡(x)+tan⁡(x))15(cos⁡3(x)+1)cos⁡2(x)\frac{16 \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{15} \left(\cos^{3}{\left(x \right)} + 1\right)}{\cos^{2}{\left(x \right)}}

Respuesta:

16(sin⁡(x)+tan⁡(x))15(cos⁡3(x)+1)cos⁡2(x)\frac{16 \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{15} \left(\cos^{3}{\left(x \right)} + 1\right)}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-5e265e26
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Primera derivada [src] 
                 15 /           2               \
(sin(x) + tan(x))  *\16 + 16*tan (x) + 16*cos(x)/
(sin⁡(x)+tan⁡(x))15(16cos⁡(x)+16tan⁡2(x)+16)\left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{15} \left(16 \cos{\left(x \right)} + 16 \tan^{2}{\left(x \right)} + 16\right)
Simplificar
Segunda derivada [src] 
                       /                         2                                                       \
                    14 |   /       2            \    /            /       2   \       \                  |
16*(sin(x) + tan(x))  *\15*\1 + tan (x) + cos(x)/  + \-sin(x) + 2*\1 + tan (x)/*tan(x)/*(sin(x) + tan(x))/
16((2(tan⁡2(x)+1)tan⁡(x)−sin⁡(x))(sin⁡(x)+tan⁡(x))+15(cos⁡(x)+tan⁡2(x)+1)2)(sin⁡(x)+tan⁡(x))1416 \left(\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right) + 15 \left(\cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)^{2}\right) \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{14}
Simplificar
Tercera derivada [src] 
                       /                          3                      /                         2                          \                                                                                 \
                    13 |    /       2            \                     2 |            /       2   \         2    /       2   \|      /            /       2   \       \                   /       2            \|
16*(sin(x) + tan(x))  *\210*\1 + tan (x) + cos(x)/  + (sin(x) + tan(x)) *\-cos(x) + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)// + 45*\-sin(x) + 2*\1 + tan (x)/*tan(x)/*(sin(x) + tan(x))*\1 + tan (x) + cos(x)//
16(sin⁡(x)+tan⁡(x))13(45(2(tan⁡2(x)+1)tan⁡(x)−sin⁡(x))(sin⁡(x)+tan⁡(x))(cos⁡(x)+tan⁡2(x)+1)+(sin⁡(x)+tan⁡(x))2(2(tan⁡2(x)+1)2+4(tan⁡2(x)+1)tan⁡2(x)−cos⁡(x))+210(cos⁡(x)+tan⁡2(x)+1)3)16 \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{13} \left(45 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right) \left(\cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) + \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)^{2} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \cos{\left(x \right)}\right) + 210 \left(\cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)^{3}\right)
Simplificar
Gráfico
Derivada de y=(sinx+tgx)^16
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