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xsinx/cosx
Derivada de la función/ xsinx/ x/cosx/ xsinx/cosx

Derivada de xsinx/cosx

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src] 
x*sin(x)
--------
 cos(x)
xsin⁡(x)cos⁡(x)\frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} 
(x*sin(x))/cos(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)=xsin⁡(x)f{\left(x \right)} = x \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. 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)=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)}

    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:

    xsin⁡2(x)+(xcos⁡(x)+sin⁡(x))cos⁡(x)cos⁡2(x)\frac{x \sin^{2}{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-1000010000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                         2   
x*cos(x) + sin(x)   x*sin (x)
----------------- + ---------
      cos(x)            2    
                     cos (x)
xsin⁡2(x)cos⁡2(x)+xcos⁡(x)+sin⁡(x)cos⁡(x)\frac{x \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{\cos{\left(x \right)}}
Simplificar
Segunda derivada [src] 
                        /         2   \                                      
                        |    2*sin (x)|          2*(x*cos(x) + sin(x))*sin(x)
2*cos(x) - x*sin(x) + x*|1 + ---------|*sin(x) + ----------------------------
                        |        2    |                     cos(x)           
                        \     cos (x) /                                      
-----------------------------------------------------------------------------
                                    cos(x)
x(2sin⁡2(x)cos⁡2(x)+1)sin⁡(x)−xsin⁡(x)+2(xcos⁡(x)+sin⁡(x))sin⁡(x)cos⁡(x)+2cos⁡(x)cos⁡(x)\frac{x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)} - x \sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \cos{\left(x \right)}}{\cos{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                                                                           /         2   \
                                                                                                      2    |    6*sin (x)|
                                                                                                 x*sin (x)*|5 + ---------|
                         /         2   \                                                                   |        2    |
                         |    2*sin (x)|                       3*(-2*cos(x) + x*sin(x))*sin(x)             \     cos (x) /
-3*sin(x) - x*cos(x) + 3*|1 + ---------|*(x*cos(x) + sin(x)) - ------------------------------- + -------------------------
                         |        2    |                                    cos(x)                         cos(x)         
                         \     cos (x) /                                                                                  
--------------------------------------------------------------------------------------------------------------------------
                                                          cos(x)
x(6sin⁡2(x)cos⁡2(x)+5)sin⁡2(x)cos⁡(x)−xcos⁡(x)−3(xsin⁡(x)−2cos⁡(x))sin⁡(x)cos⁡(x)+3(xcos⁡(x)+sin⁡(x))(2sin⁡2(x)cos⁡2(x)+1)−3sin⁡(x)cos⁡(x)\frac{\frac{x \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos{\left(x \right)}} - x \cos{\left(x \right)} - \frac{3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) - 3 \sin{\left(x \right)}}{\cos{\left(x \right)}}
Simplificar
Gráfico
Derivada de xsinx/cosx
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