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y=lnx*(ctgx(x)*cos(x))
Derivada de la función/ cos(x)/ y=lnx/ y=lnx*(ctgx(x)*cos(x))

Derivada de y=lnx*(ctgx(x)*cos(x))

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

Ha introducido [src] 
log(x)*cot(x)*x*cos(x)
xcot⁡(x)cos⁡(x)log⁡(x)x \cot{\left(x \right)} \cos{\left(x \right)} \log{\left(x \right)} 
log(x)*((cot(x)*x)*cos(x))
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)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

    g(x)=xcot⁡(x)cos⁡(x)g{\left(x \right)} = x \cot{\left(x \right)} \cos{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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)=xcot⁡(x)f{\left(x \right)} = x \cot{\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)=cot⁡(x)f{\left(x \right)} = \cot{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\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)}}

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

        1. Según el principio, aplicamos: xx tenemos 11

        Como resultado de: −x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)+cot⁡(x)- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \cot{\left(x \right)}

      g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; calculamos 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)}

      Como resultado de: −xsin⁡(x)cot⁡(x)+(−x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)+cot⁡(x))cos⁡(x)- x \sin{\left(x \right)} \cot{\left(x \right)} + \left(- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \cot{\left(x \right)}\right) \cos{\left(x \right)}

    Como resultado de: (−xsin⁡(x)cot⁡(x)+(−x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)+cot⁡(x))cos⁡(x))log⁡(x)+cos⁡(x)cot⁡(x)\left(- x \sin{\left(x \right)} \cot{\left(x \right)} + \left(- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \cot{\left(x \right)}\right) \cos{\left(x \right)}\right) \log{\left(x \right)} + \cos{\left(x \right)} \cot{\left(x \right)}

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-5000050000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
//  /        2   \         \                         \                       
\\x*\-1 - cot (x)/ + cot(x)/*cos(x) - x*cot(x)*sin(x)/*log(x) + cos(x)*cot(x)
(−xsin⁡(x)cot⁡(x)+(x(−cot⁡2(x)−1)+cot⁡(x))cos⁡(x))log⁡(x)+cos⁡(x)cot⁡(x)\left(- x \sin{\left(x \right)} \cot{\left(x \right)} + \left(x \left(- \cot^{2}{\left(x \right)} - 1\right) + \cot{\left(x \right)}\right) \cos{\left(x \right)}\right) \log{\left(x \right)} + \cos{\left(x \right)} \cot{\left(x \right)}
Simplificar
Segunda derivada [src] 
 /                                                                                                                        //            /       2   \\                         \                \
 |/    /            /       2   \\            /       2        /       2   \       \                         \          2*\\-cot(x) + x*\1 + cot (x)//*cos(x) + x*cot(x)*sin(x)/   cos(x)*cot(x)|
-|\- 2*\-cot(x) + x*\1 + cot (x)//*sin(x) + 2*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/*cos(x) + x*cos(x)*cot(x)/*log(x) + -------------------------------------------------------- + -------------|
 \                                                                                                                                                 x                                     x      /
−((xcos⁡(x)cot⁡(x)−2(x(cot⁡2(x)+1)−cot⁡(x))sin⁡(x)+2(−x(cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1)cos⁡(x))log⁡(x)+2(xsin⁡(x)cot⁡(x)+(x(cot⁡2(x)+1)−cot⁡(x))cos⁡(x))x+cos⁡(x)cot⁡(x)x)- (\left(x \cos{\left(x \right)} \cot{\left(x \right)} - 2 \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right) \sin{\left(x \right)} + 2 \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}\right) \log{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} \cot{\left(x \right)} + \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right) \cos{\left(x \right)}\right)}{x} + \frac{\cos{\left(x \right)} \cot{\left(x \right)}}{x})
Simplificar
Tercera derivada [src] 
                                                                                                                                                                               /    /            /       2   \\            /       2        /       2   \       \                         \     //            /       2   \\                         \                  
/  /            /       2   \\            /       2        /       2   \       \                              /       2   \ /              /         2   \\       \          3*\- 2*\-cot(x) + x*\1 + cot (x)//*sin(x) + 2*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/*cos(x) + x*cos(x)*cot(x)/   3*\\-cot(x) + x*\1 + cot (x)//*cos(x) + x*cot(x)*sin(x)/   2*cos(x)*cot(x)
\3*\-cot(x) + x*\1 + cot (x)//*cos(x) + 6*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/*sin(x) + x*cot(x)*sin(x) - 2*\1 + cot (x)/*\-3*cot(x) + x*\1 + 3*cot (x)//*cos(x)/*log(x) - -------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------- + ---------------
                                                                                                                                                                                                                                   x                                                                                      2                                      2      
                                                                                                                                                                                                                                                                                                                         x                                      x
(xsin⁡(x)cot⁡(x)+3(x(cot⁡2(x)+1)−cot⁡(x))cos⁡(x)−2(x(3cot⁡2(x)+1)−3cot⁡(x))(cot⁡2(x)+1)cos⁡(x)+6(−x(cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1)sin⁡(x))log⁡(x)−3(xcos⁡(x)cot⁡(x)−2(x(cot⁡2(x)+1)−cot⁡(x))sin⁡(x)+2(−x(cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1)cos⁡(x))x+3(xsin⁡(x)cot⁡(x)+(x(cot⁡2(x)+1)−cot⁡(x))cos⁡(x))x2+2cos⁡(x)cot⁡(x)x2\left(x \sin{\left(x \right)} \cot{\left(x \right)} + 3 \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right) \cos{\left(x \right)} - 2 \left(x \left(3 \cot^{2}{\left(x \right)} + 1\right) - 3 \cot{\left(x \right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} + 6 \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}\right) \log{\left(x \right)} - \frac{3 \left(x \cos{\left(x \right)} \cot{\left(x \right)} - 2 \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right) \sin{\left(x \right)} + 2 \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}\right)}{x} + \frac{3 \left(x \sin{\left(x \right)} \cot{\left(x \right)} + \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right) \cos{\left(x \right)}\right)}{x^{2}} + \frac{2 \cos{\left(x \right)} \cot{\left(x \right)}}{x^{2}}
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Gráfico
Derivada de y=lnx*(ctgx(x)*cos(x))
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