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-x*sin(x)+tg(x)*cos(x)*ctg(x)
Derivada de la función/ -x*sin(x)+tg(x)*cos(x)*ctg(x)

Derivada de -x*sin(x)+tg(x)*cos(x)*ctg(x)

Función f() - derivada -er orden en el punto
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Ha introducido [src] 
-x*sin(x) + tan(x)*cos(x)*cot(x)
−xsin⁡(x)+cos⁡(x)tan⁡(x)cot⁡(x)- x \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)} \cot{\left(x \right)} 
(-x)*sin(x) + (tan(x)*cos(x))*cot(x)
Solución detallada

  1. diferenciamos −xsin⁡(x)+cos⁡(x)tan⁡(x)cot⁡(x)- x \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)} \cot{\left(x \right)} miembro por miembro:

    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. 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

      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: −xcos⁡(x)−sin⁡(x)- x \cos{\left(x \right)} - \sin{\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)=cos⁡(x)tan⁡(x)f{\left(x \right)} = \cos{\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:

          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)}}

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

      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. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\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: (sin⁡2(x)+cos⁡2(x)cos⁡(x)−sin⁡(x)tan⁡(x))cot⁡(x)−sin⁡2(x)+cos⁡2(x)cos⁡(x)tan⁡(x)\left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan{\left(x \right)}\right) \cot{\left(x \right)} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}}

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-2020
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
          //       2   \                       \                     /        2   \              
-sin(x) + \\1 + tan (x)/*cos(x) - sin(x)*tan(x)/*cot(x) - x*cos(x) + \-1 - cot (x)/*cos(x)*tan(x)
−xcos⁡(x)+((tan⁡2(x)+1)cos⁡(x)−sin⁡(x)tan⁡(x))cot⁡(x)+(−cot⁡2(x)−1)cos⁡(x)tan⁡(x)−sin⁡(x)- x \cos{\left(x \right)} + \left(\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}\right) \cot{\left(x \right)} + \left(- \cot^{2}{\left(x \right)} - 1\right) \cos{\left(x \right)} \tan{\left(x \right)} - \sin{\left(x \right)}
Simplificar
Segunda derivada [src] 
                       /       2   \ //       2   \                       \   /                  /       2   \            /       2   \              \          /       2   \                 /       2   \ /       2   \            /       2   \                     
-2*cos(x) + x*sin(x) - \1 + cot (x)/*\\1 + tan (x)/*cos(x) - sin(x)*tan(x)/ - \cos(x)*tan(x) + 2*\1 + tan (x)/*sin(x) - 2*\1 + tan (x)/*cos(x)*tan(x)/*cot(x) + \1 + cot (x)/*sin(x)*tan(x) - \1 + cot (x)/*\1 + tan (x)/*cos(x) + 2*\1 + cot (x)/*cos(x)*cot(x)*tan(x)
xsin⁡(x)−((tan⁡2(x)+1)cos⁡(x)−sin⁡(x)tan⁡(x))(cot⁡2(x)+1)−(tan⁡2(x)+1)(cot⁡2(x)+1)cos⁡(x)+(cot⁡2(x)+1)sin⁡(x)tan⁡(x)+2(cot⁡2(x)+1)cos⁡(x)tan⁡(x)cot⁡(x)−(2(tan⁡2(x)+1)sin⁡(x)−2(tan⁡2(x)+1)cos⁡(x)tan⁡(x)+cos⁡(x)tan⁡(x))cot⁡(x)−2cos⁡(x)x \sin{\left(x \right)} - \left(\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right) - \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} \cot{\left(x \right)} - \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 2 \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) \cot{\left(x \right)} - 2 \cos{\left(x \right)}
Simplificar
Tercera derivada [src] 
                      /                                                        2                                                                        \                                                                                                                                                  2                                                                                                                                                                                                                                                                                                
                      |                  /       2   \            /       2   \             /       2   \                      2    /       2   \       |            /       2   \ /                  /       2   \            /       2   \              \   /       2   \                   /       2   \                    /       2   \ /       2   \            /       2   \ //       2   \                       \               2    /       2   \                   /       2   \                          /       2   \ /       2   \                   /       2   \ /       2   \              
3*sin(x) + x*cos(x) + \sin(x)*tan(x) - 3*\1 + tan (x)/*cos(x) + 2*\1 + tan (x)/ *cos(x) - 6*\1 + tan (x)/*sin(x)*tan(x) + 4*tan (x)*\1 + tan (x)/*cos(x)/*cot(x) + 2*\1 + cot (x)/*\cos(x)*tan(x) + 2*\1 + tan (x)/*sin(x) - 2*\1 + tan (x)/*cos(x)*tan(x)/ + \1 + cot (x)/*cos(x)*tan(x) - 2*\1 + cot (x)/ *cos(x)*tan(x) + 2*\1 + cot (x)/*\1 + tan (x)/*sin(x) + 2*\1 + cot (x)/*\\1 + tan (x)/*cos(x) - sin(x)*tan(x)/*cot(x) - 4*cot (x)*\1 + cot (x)/*cos(x)*tan(x) - 4*\1 + cot (x)/*cot(x)*sin(x)*tan(x) - 2*\1 + cot (x)/*\1 + tan (x)/*cos(x)*tan(x) + 4*\1 + cot (x)/*\1 + tan (x)/*cos(x)*cot(x)
xcos⁡(x)+2((tan⁡2(x)+1)cos⁡(x)−sin⁡(x)tan⁡(x))(cot⁡2(x)+1)cot⁡(x)+2(tan⁡2(x)+1)(cot⁡2(x)+1)sin⁡(x)−2(tan⁡2(x)+1)(cot⁡2(x)+1)cos⁡(x)tan⁡(x)+4(tan⁡2(x)+1)(cot⁡2(x)+1)cos⁡(x)cot⁡(x)−2(cot⁡2(x)+1)2cos⁡(x)tan⁡(x)+2(cot⁡2(x)+1)(2(tan⁡2(x)+1)sin⁡(x)−2(tan⁡2(x)+1)cos⁡(x)tan⁡(x)+cos⁡(x)tan⁡(x))−4(cot⁡2(x)+1)sin⁡(x)tan⁡(x)cot⁡(x)−4(cot⁡2(x)+1)cos⁡(x)tan⁡(x)cot⁡2(x)+(cot⁡2(x)+1)cos⁡(x)tan⁡(x)+(2(tan⁡2(x)+1)2cos⁡(x)−6(tan⁡2(x)+1)sin⁡(x)tan⁡(x)+4(tan⁡2(x)+1)cos⁡(x)tan⁡2(x)−3(tan⁡2(x)+1)cos⁡(x)+sin⁡(x)tan⁡(x))cot⁡(x)+3sin⁡(x)x \cos{\left(x \right)} + 2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \cot{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)} \tan{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 2 \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) - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} \cot{\left(x \right)} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} \cot^{2}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} + \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan^{2}{\left(x \right)} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} + \sin{\left(x \right)} \tan{\left(x \right)}\right) \cot{\left(x \right)} + 3 \sin{\left(x \right)}
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Gráfico
Derivada de -x*sin(x)+tg(x)*cos(x)*ctg(x)
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