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y=x^3ctgx(x+1/2^x)
Derivada de la función/ 3ctgx/ y=x^3/ x+1/2/ 1/2^x/ y=x^3ctgx(x+1/2^x)

Derivada de y=x^3ctgx(x+1/2^x)

Función f() - derivada -er orden en el punto
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Solución

Ha introducido [src] 
 3        /     -x\
x *cot(x)*\x + 2  /
x3cot⁡(x)(x+(12)x)x^{3} \cot{\left(x \right)} \left(x + \left(\frac{1}{2}\right)^{x}\right) 
(x^3*cot(x))*(x + (1/2)^x)
Solución detallada

  1. 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)=x3(2xx+1)cot⁡(x)f{\left(x \right)} = x^{3} \left(2^{x} x + 1\right) \cot{\left(x \right)} y g(x)=2xg{\left(x \right)} = 2^{x}.

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

      f(x)=x3f{\left(x \right)} = x^{3}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      g(x)=2xx+1g{\left(x \right)} = 2^{x} x + 1; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. diferenciamos 2xx+12^{x} x + 1 miembro por miembro:

        1. La derivada de una constante 11 es igual a cero.

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

          1. Según el principio, aplicamos: xx tenemos 11

          g(x)=2xg{\left(x \right)} = 2^{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

          1. ddx2x=2xlog⁡(2)\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}

          Como resultado de: 2xxlog⁡(2)+2x2^{x} x \log{\left(2 \right)} + 2^{x}

        Como resultado de: 2xxlog⁡(2)+2x2^{x} x \log{\left(2 \right)} + 2^{x}

      h(x)=cot⁡(x)h{\left(x \right)} = \cot{\left(x \right)}; calculamos ddxh(x)\frac{d}{d x} h{\left(x \right)}:

      1. Hay varias formas de calcular esta derivada.

        Method #1

        1. Reescribimos las funciones para diferenciar:

          cot⁡(x)=1tan⁡(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

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

        3. Según el principio, aplicamos: 1u\frac{1}{u} tenemos −1u2- \frac{1}{u^{2}}

        4. 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:

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

        Method #2

        1. Reescribimos las funciones para diferenciar:

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

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

          Para calcular 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)}

          Ahora aplicamos la regla de la derivada de una divesión:

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

      Como resultado de: −x3(2xx+1)(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)+x3(2xxlog⁡(2)+2x)cot⁡(x)+3x2(2xx+1)cot⁡(x)- \frac{x^{3} \left(2^{x} x + 1\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + x^{3} \left(2^{x} x \log{\left(2 \right)} + 2^{x}\right) \cot{\left(x \right)} + 3 x^{2} \left(2^{x} x + 1\right) \cot{\left(x \right)}

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

    1. ddx2x=2xlog⁡(2)\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    2−2x(−2xx3(2xx+1)log⁡(2)cot⁡(x)+2x(−x3(2xx+1)(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)+x3(2xxlog⁡(2)+2x)cot⁡(x)+3x2(2xx+1)cot⁡(x)))2^{- 2 x} \left(- 2^{x} x^{3} \left(2^{x} x + 1\right) \log{\left(2 \right)} \cot{\left(x \right)} + 2^{x} \left(- \frac{x^{3} \left(2^{x} x + 1\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + x^{3} \left(2^{x} x \log{\left(2 \right)} + 2^{x}\right) \cot{\left(x \right)} + 3 x^{2} \left(2^{x} x + 1\right) \cot{\left(x \right)}\right)\right)

  2. Simplificamos:

    2−xx2(−2x+1x2+2x+2xsin⁡(2x)−xlog⁡(2)sin⁡(2x)−2x+3sin⁡(2x))1−cos⁡(2x)\frac{2^{- x} x^{2} \left(- 2^{x + 1} x^{2} + 2^{x + 2} x \sin{\left(2 x \right)} - x \log{\left(2 \right)} \sin{\left(2 x \right)} - 2 x + 3 \sin{\left(2 x \right)}\right)}{1 - \cos{\left(2 x \right)}}

Respuesta:

2−xx2(−2x+1x2+2x+2xsin⁡(2x)−xlog⁡(2)sin⁡(2x)−2x+3sin⁡(2x))1−cos⁡(2x)\frac{2^{- x} x^{2} \left(- 2^{x + 1} x^{2} + 2^{x + 2} x \sin{\left(2 x \right)} - x \log{\left(2 \right)} \sin{\left(2 x \right)} - 2 x + 3 \sin{\left(2 x \right)}\right)}{1 - \cos{\left(2 x \right)}}

Gráfica
02468-8-6-4-2-1010-5000000001000000000
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Trazar el gráfico f'(x)
Primera derivada [src] 
/     -x\ / 3 /        2   \      2       \    3 /     -x       \       
\x + 2  /*\x *\-1 - cot (x)/ + 3*x *cot(x)/ + x *\1 - 2  *log(2)/*cot(x)
x3(1−2−xlog⁡(2))cot⁡(x)+(x+(12)x)(x3(−cot⁡2(x)−1)+3x2cot⁡(x))x^{3} \left(1 - 2^{- x} \log{\left(2 \right)}\right) \cot{\left(x \right)} + \left(x + \left(\frac{1}{2}\right)^{x}\right) \left(x^{3} \left(- \cot^{2}{\left(x \right)} - 1\right) + 3 x^{2} \cot{\left(x \right)}\right)
Simplificar
Segunda derivada [src] 
  /  /     -x\ /               /       2   \    2 /       2   \       \       /     -x       \ /              /       2   \\    -x  2    2          \
x*\2*\x + 2  /*\3*cot(x) - 3*x*\1 + cot (x)/ + x *\1 + cot (x)/*cot(x)/ - 2*x*\1 - 2  *log(2)/*\-3*cot(x) + x*\1 + cot (x)// + 2  *x *log (2)*cot(x)/
x(−2x(1−2−xlog⁡(2))(x(cot⁡2(x)+1)−3cot⁡(x))+2(x+2−x)(x2(cot⁡2(x)+1)cot⁡(x)−3x(cot⁡2(x)+1)+3cot⁡(x))+2−xx2log⁡(2)2cot⁡(x))x \left(- 2 x \left(1 - 2^{- x} \log{\left(2 \right)}\right) \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - 3 \cot{\left(x \right)}\right) + 2 \left(x + 2^{- x}\right) \left(x^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 3 x \left(\cot^{2}{\left(x \right)} + 1\right) + 3 \cot{\left(x \right)}\right) + 2^{- x} x^{2} \log{\left(2 \right)}^{2} \cot{\left(x \right)}\right)
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3-я производная [src] 
    /     -x\ /                /       2   \    3 /       2   \ /         2   \      2 /       2   \       \       /     -x       \ /               /       2   \    2 /       2   \       \    -x  3    3                -x  2    2    /              /       2   \\
- 2*\x + 2  /*\-3*cot(x) + 9*x*\1 + cot (x)/ + x *\1 + cot (x)/*\1 + 3*cot (x)/ - 9*x *\1 + cot (x)/*cot(x)/ + 6*x*\1 - 2  *log(2)/*\3*cot(x) - 3*x*\1 + cot (x)/ + x *\1 + cot (x)/*cot(x)/ - 2  *x *log (2)*cot(x) - 3*2  *x *log (2)*\-3*cot(x) + x*\1 + cot (x)//
6x(1−2−xlog⁡(2))(x2(cot⁡2(x)+1)cot⁡(x)−3x(cot⁡2(x)+1)+3cot⁡(x))−2(x+2−x)(x3(cot⁡2(x)+1)(3cot⁡2(x)+1)−9x2(cot⁡2(x)+1)cot⁡(x)+9x(cot⁡2(x)+1)−3cot⁡(x))−2−xx3log⁡(2)3cot⁡(x)−3⋅2−xx2(x(cot⁡2(x)+1)−3cot⁡(x))log⁡(2)26 x \left(1 - 2^{- x} \log{\left(2 \right)}\right) \left(x^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 3 x \left(\cot^{2}{\left(x \right)} + 1\right) + 3 \cot{\left(x \right)}\right) - 2 \left(x + 2^{- x}\right) \left(x^{3} \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) - 9 x^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + 9 x \left(\cot^{2}{\left(x \right)} + 1\right) - 3 \cot{\left(x \right)}\right) - 2^{- x} x^{3} \log{\left(2 \right)}^{3} \cot{\left(x \right)} - 3 \cdot 2^{- x} x^{2} \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - 3 \cot{\left(x \right)}\right) \log{\left(2 \right)}^{2}
Simplificar
Tercera derivada [src] 
    /     -x\ /                /       2   \    3 /       2   \ /         2   \      2 /       2   \       \       /     -x       \ /               /       2   \    2 /       2   \       \    -x  3    3                -x  2    2    /              /       2   \\
- 2*\x + 2  /*\-3*cot(x) + 9*x*\1 + cot (x)/ + x *\1 + cot (x)/*\1 + 3*cot (x)/ - 9*x *\1 + cot (x)/*cot(x)/ + 6*x*\1 - 2  *log(2)/*\3*cot(x) - 3*x*\1 + cot (x)/ + x *\1 + cot (x)/*cot(x)/ - 2  *x *log (2)*cot(x) - 3*2  *x *log (2)*\-3*cot(x) + x*\1 + cot (x)//
6x(1−2−xlog⁡(2))(x2(cot⁡2(x)+1)cot⁡(x)−3x(cot⁡2(x)+1)+3cot⁡(x))−2(x+2−x)(x3(cot⁡2(x)+1)(3cot⁡2(x)+1)−9x2(cot⁡2(x)+1)cot⁡(x)+9x(cot⁡2(x)+1)−3cot⁡(x))−2−xx3log⁡(2)3cot⁡(x)−3⋅2−xx2(x(cot⁡2(x)+1)−3cot⁡(x))log⁡(2)26 x \left(1 - 2^{- x} \log{\left(2 \right)}\right) \left(x^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 3 x \left(\cot^{2}{\left(x \right)} + 1\right) + 3 \cot{\left(x \right)}\right) - 2 \left(x + 2^{- x}\right) \left(x^{3} \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) - 9 x^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + 9 x \left(\cot^{2}{\left(x \right)} + 1\right) - 3 \cot{\left(x \right)}\right) - 2^{- x} x^{3} \log{\left(2 \right)}^{3} \cot{\left(x \right)} - 3 \cdot 2^{- x} x^{2} \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - 3 \cot{\left(x \right)}\right) \log{\left(2 \right)}^{2}
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Gráfico
Derivada de y=x^3ctgx(x+1/2^x)
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