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y=x((lnx)/(sinx))+x(ctgx)
Derivada de la función/ (sinx)/ y=x((lnx)/(sinx))+x(ctgx)

Derivada de y=x((lnx)/(sinx))+x(ctgx)

Función f() - derivada -er orden en el punto
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Ha introducido [src] 
  log(x)           
x*------ + x*cot(x)
  sin(x)
xlog⁡(x)sin⁡(x)+xcot⁡(x)x \frac{\log{\left(x \right)}}{\sin{\left(x \right)}} + x \cot{\left(x \right)} 
x*(log(x)/sin(x)) + x*cot(x)
Solución detallada

  1. diferenciamos xlog⁡(x)sin⁡(x)+xcot⁡(x)x \frac{\log{\left(x \right)}}{\sin{\left(x \right)}} + x \cot{\left(x \right)} miembro por miembro:

    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)=xlog⁡(x)f{\left(x \right)} = x \log{\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. 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)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

        Como resultado de: log⁡(x)+1\log{\left(x \right)} + 1

      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:

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

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

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

  2. Simplificamos:

    −2xlog⁡(x)cos⁡(x)−2x+2log⁡(x)sin⁡(x)+2sin⁡(x)+sin⁡(2x)1−cos⁡(2x)\frac{- 2 x \log{\left(x \right)} \cos{\left(x \right)} - 2 x + 2 \log{\left(x \right)} \sin{\left(x \right)} + 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{1 - \cos{\left(2 x \right)}}

Respuesta:

−2xlog⁡(x)cos⁡(x)−2x+2log⁡(x)sin⁡(x)+2sin⁡(x)+sin⁡(2x)1−cos⁡(2x)\frac{- 2 x \log{\left(x \right)} \cos{\left(x \right)} - 2 x + 2 \log{\left(x \right)} \sin{\left(x \right)} + 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{1 - \cos{\left(2 x \right)}}

Gráfica
02468-8-6-4-2-1010-2000020000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
  /        2   \     /   1       cos(x)*log(x)\   log(x)         
x*\-1 - cot (x)/ + x*|-------- - -------------| + ------ + cot(x)
                     |x*sin(x)         2      |   sin(x)         
                     \              sin (x)   /
x(−log⁡(x)cos⁡(x)sin⁡2(x)+1xsin⁡(x))+x(−cot⁡2(x)−1)+log⁡(x)sin⁡(x)+cot⁡(x)x \left(- \frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x \sin{\left(x \right)}}\right) + x \left(- \cot^{2}{\left(x \right)} - 1\right) + \frac{\log{\left(x \right)}}{\sin{\left(x \right)}} + \cot{\left(x \right)}
Simplificar
Segunda derivada [src] 
                              /                       2                   \                                             
                              |  1    2*cos(x)   2*cos (x)*log(x)         |                                             
                            x*|- -- - -------- + ---------------- + log(x)|                                             
                              |   2   x*sin(x)          2                 |                                             
          2         2         \  x                   sin (x)              /   2*cos(x)*log(x)       /       2   \       
-2 - 2*cot (x) + -------- + ----------------------------------------------- - --------------- + 2*x*\1 + cot (x)/*cot(x)
                 x*sin(x)                        sin(x)                              2                                  
                                                                                  sin (x)
2x(cot⁡2(x)+1)cot⁡(x)+x(log⁡(x)+2log⁡(x)cos⁡2(x)sin⁡2(x)−2cos⁡(x)xsin⁡(x)−1x2)sin⁡(x)−2log⁡(x)cos⁡(x)sin⁡2(x)−2cot⁡2(x)−2+2xsin⁡(x)2 x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{x \left(\log{\left(x \right)} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{1}{x^{2}}\right)}{\sin{\left(x \right)}} - \frac{2 \log{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 2 \cot^{2}{\left(x \right)} - 2 + \frac{2}{x \sin{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                                         /              3                                                2   \                                                           
                                                                         |2    3   6*cos (x)*log(x)   5*cos(x)*log(x)    3*cos(x)   6*cos (x)|                                                           
                                                                       x*|-- + - - ---------------- - --------------- + --------- + ---------|                                                           
                               2                                         | 3   x          3                sin(x)        2               2   |                                                2          
      3           /       2   \    3*log(x)     /       2   \            \x            sin (x)                          x *sin(x)   x*sin (x)/    6*cos(x)          2    /       2   \   6*cos (x)*log(x)
- --------- - 2*x*\1 + cot (x)/  + -------- + 6*\1 + cot (x)/*cot(x) + ----------------------------------------------------------------------- - --------- - 4*x*cot (x)*\1 + cot (x)/ + ----------------
   2                                sin(x)                                                              sin(x)                                        2                                         3        
  x *sin(x)                                                                                                                                      x*sin (x)                                   sin (x)
−2x(cot⁡2(x)+1)2−4x(cot⁡2(x)+1)cot⁡2(x)+x(−5log⁡(x)cos⁡(x)sin⁡(x)−6log⁡(x)cos⁡3(x)sin⁡3(x)+3x+6cos⁡2(x)xsin⁡2(x)+3cos⁡(x)x2sin⁡(x)+2x3)sin⁡(x)+6(cot⁡2(x)+1)cot⁡(x)+3log⁡(x)sin⁡(x)+6log⁡(x)cos⁡2(x)sin⁡3(x)−6cos⁡(x)xsin⁡2(x)−3x2sin⁡(x)- 2 x \left(\cot^{2}{\left(x \right)} + 1\right)^{2} - 4 x \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \frac{x \left(- \frac{5 \log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{6 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{3}{x} + \frac{6 \cos^{2}{\left(x \right)}}{x \sin^{2}{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2}{x^{3}}\right)}{\sin{\left(x \right)}} + 6 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{3 \log{\left(x \right)}}{\sin{\left(x \right)}} + \frac{6 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{6 \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{3}{x^{2} \sin{\left(x \right)}}
Simplificar
Gráfico
Derivada de y=x((lnx)/(sinx))+x(ctgx)
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