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y=tg(x)*sin(x)*(x^3-x+2)

Derivada de y=tg(x)*sin(x)*(x^3-x+2)

Función f() - derivada -er orden en el punto
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Ha introducido [src]
              / 3        \
tan(x)*sin(x)*\x  - x + 2/
sin(x)tan(x)((x3x)+2)\sin{\left(x \right)} \tan{\left(x \right)} \left(\left(x^{3} - x\right) + 2\right)
(tan(x)*sin(x))*(x^3 - x + 2)
Solución detallada
  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=sin(x)tan(x)f{\left(x \right)} = \sin{\left(x \right)} \tan{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=tan(x)f{\left(x \right)} = \tan{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\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:

        sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      g(x)=sin(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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: (sin2(x)+cos2(x))sin(x)cos2(x)+cos(x)tan(x)\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \cos{\left(x \right)} \tan{\left(x \right)}

    g(x)=(x3x)+2g{\left(x \right)} = \left(x^{3} - x\right) + 2; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. diferenciamos (x3x)+2\left(x^{3} - x\right) + 2 miembro por miembro:

      1. diferenciamos x3xx^{3} - x miembro por miembro:

        1. Según el principio, aplicamos: x3x^{3} tenemos 3x23 x^{2}

        2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

          1. Según el principio, aplicamos: xx tenemos 11

          Entonces, como resultado: 1-1

        Como resultado de: 3x213 x^{2} - 1

      2. La derivada de una constante 22 es igual a cero.

      Como resultado de: 3x213 x^{2} - 1

    Como resultado de: (3x21)sin(x)tan(x)+((sin2(x)+cos2(x))sin(x)cos2(x)+cos(x)tan(x))((x3x)+2)\left(3 x^{2} - 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + \left(\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \cos{\left(x \right)} \tan{\left(x \right)}\right) \left(\left(x^{3} - x\right) + 2\right)

  2. Simplificamos:

    ((2sin2(x))(x3x+2)+(3x21)sin(x)cos(x))sin(x)cos2(x)\frac{\left(\left(2 - \sin^{2}{\left(x \right)}\right) \left(x^{3} - x + 2\right) + \left(3 x^{2} - 1\right) \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}


Respuesta:

((2sin2(x))(x3x+2)+(3x21)sin(x)cos(x))sin(x)cos2(x)\frac{\left(\left(2 - \sin^{2}{\left(x \right)}\right) \left(x^{3} - x + 2\right) + \left(3 x^{2} - 1\right) \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-200000200000
Primera derivada [src]
//       2   \                       \ / 3        \   /        2\              
\\1 + tan (x)/*sin(x) + cos(x)*tan(x)/*\x  - x + 2/ + \-1 + 3*x /*sin(x)*tan(x)
(3x21)sin(x)tan(x)+((tan2(x)+1)sin(x)+cos(x)tan(x))((x3x)+2)\left(3 x^{2} - 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + \left(\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}\right) \left(\left(x^{3} - x\right) + 2\right)
Segunda derivada [src]
/     3    \ /                   /       2   \            /       2   \              \     /        2\ //       2   \                       \                    
\2 + x  - x/*\-sin(x)*tan(x) + 2*\1 + tan (x)/*cos(x) + 2*\1 + tan (x)/*sin(x)*tan(x)/ + 2*\-1 + 3*x /*\\1 + tan (x)/*sin(x) + cos(x)*tan(x)/ + 6*x*sin(x)*tan(x)
6xsin(x)tan(x)+2(3x21)((tan2(x)+1)sin(x)+cos(x)tan(x))+(x3x+2)(2(tan2(x)+1)sin(x)tan(x)+2(tan2(x)+1)cos(x)sin(x)tan(x))6 x \sin{\left(x \right)} \tan{\left(x \right)} + 2 \left(3 x^{2} - 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}\right) + \left(x^{3} - x + 2\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}\right)
Tercera derivada [src]
/     3    \ /                   /       2   \            /       2   \ /         2   \            /       2   \              \     /        2\ /                   /       2   \            /       2   \              \                          //       2   \                       \
\2 + x  - x/*\-cos(x)*tan(x) - 3*\1 + tan (x)/*sin(x) + 2*\1 + tan (x)/*\1 + 3*tan (x)/*sin(x) + 6*\1 + tan (x)/*cos(x)*tan(x)/ + 3*\-1 + 3*x /*\-sin(x)*tan(x) + 2*\1 + tan (x)/*cos(x) + 2*\1 + tan (x)/*sin(x)*tan(x)/ + 6*sin(x)*tan(x) + 18*x*\\1 + tan (x)/*sin(x) + cos(x)*tan(x)/
18x((tan2(x)+1)sin(x)+cos(x)tan(x))+3(3x21)(2(tan2(x)+1)sin(x)tan(x)+2(tan2(x)+1)cos(x)sin(x)tan(x))+(x3x+2)(2(tan2(x)+1)(3tan2(x)+1)sin(x)3(tan2(x)+1)sin(x)+6(tan2(x)+1)cos(x)tan(x)cos(x)tan(x))+6sin(x)tan(x)18 x \left(\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}\right) + 3 \left(3 x^{2} - 1\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}\right) + \left(x^{3} - x + 2\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} - \cos{\left(x \right)} \tan{\left(x \right)}\right) + 6 \sin{\left(x \right)} \tan{\left(x \right)}
Gráfico
Derivada de y=tg(x)*sin(x)*(x^3-x+2)