Derivada de y=sinx/(1+lnsinx)

Ha introducido [src]

      sin(x)     
-----------------
1 + log(x)*sin(x)

\frac{\sin{\left(x \right)}}{\log{\left(x \right)} \sin{\left(x \right)} + 1}

sin(x)/(1 + log(x)*sin(x))

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} + 1$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. La derivada del seno es igual al coseno:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $\log{\left(x \right)} \sin{\left(x \right)} + 1$ miembro por miembro:
  1. La derivada de una constante $1$ es igual a cero.
  2. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
    $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de: $\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}$
  Como resultado de: $\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{\left(\log{\left(x \right)} \sin{\left(x \right)} + 1\right) \cos{\left(x \right)} - \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{\left(\log{\left(x \right)} \sin{\left(x \right)} + 1\right)^{2}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                    /  sin(x)                \       
                    |- ------ - cos(x)*log(x)|*sin(x)
      cos(x)        \    x                   /       
----------------- + ---------------------------------
1 + log(x)*sin(x)                             2      
                           (1 + log(x)*sin(x))

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

Simplificar

Segunda derivada [src]

          /                                                              2\                                           
          |                                      /sin(x)                \ |                                           
          |                                    2*|------ + cos(x)*log(x)| |                                           
          |sin(x)                   2*cos(x)     \  x                   / |                                           
          |------ + log(x)*sin(x) - -------- + ---------------------------|*sin(x)     /sin(x)                \       
          |   2                        x            1 + log(x)*sin(x)     |          2*|------ + cos(x)*log(x)|*cos(x)
          \  x                                                            /            \  x                   /       
-sin(x) + ------------------------------------------------------------------------ - ---------------------------------
                                     1 + log(x)*sin(x)                                       1 + log(x)*sin(x)        
----------------------------------------------------------------------------------------------------------------------
                                                  1 + log(x)*sin(x)

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

Simplificar

Tercera derivada [src]

          /                                          3                                      /sin(x)                \ /sin(x)                   2*cos(x)\\                                                /                                                              2\       
          |                  /sin(x)                \                                     6*|------ + cos(x)*log(x)|*|------ + log(x)*sin(x) - --------||                                                |                                      /sin(x)                \ |       
          |                6*|------ + cos(x)*log(x)|                                       \  x                   / |   2                        x    ||                                                |                                    2*|------ + cos(x)*log(x)| |       
          |                  \  x                   /    2*sin(x)   3*sin(x)   3*cos(x)                              \  x                              /|                                                |sin(x)                   2*cos(x)     \  x                   / |       
          |cos(x)*log(x) - --------------------------- - -------- + -------- + -------- - --------------------------------------------------------------|*sin(x)     /sin(x)                \          3*|------ + log(x)*sin(x) - -------- + ---------------------------|*cos(x)
          |                                       2          3         x           2                            1 + log(x)*sin(x)                       |          3*|------ + cos(x)*log(x)|*sin(x)     |   2                        x            1 + log(x)*sin(x)     |       
          \                    (1 + log(x)*sin(x))          x                     x                                                                     /            \  x                   /            \  x                                                            /       
-cos(x) + ------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------- + --------------------------------------------------------------------------
                                                                            1 + log(x)*sin(x)                                                                              1 + log(x)*sin(x)                                       1 + log(x)*sin(x)                             
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                1 + log(x)*sin(x)

\frac{- \cos{\left(x \right)} + \frac{3 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{\log{\left(x \right)} \sin{\left(x \right)} + 1} + \frac{3 \left(\log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2}}{\log{\left(x \right)} \sin{\left(x \right)} + 1} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) \cos{\left(x \right)}}{\log{\left(x \right)} \sin{\left(x \right)} + 1} + \frac{\left(\log{\left(x \right)} \cos{\left(x \right)} - \frac{6 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right)}{\log{\left(x \right)} \sin{\left(x \right)} + 1} - \frac{6 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3}}{\left(\log{\left(x \right)} \sin{\left(x \right)} + 1\right)^{2}} + \frac{3 \sin{\left(x \right)}}{x} + \frac{3 \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin{\left(x \right)}}{x^{3}}\right) \sin{\left(x \right)}}{\log{\left(x \right)} \sin{\left(x \right)} + 1}}{\log{\left(x \right)} \sin{\left(x \right)} + 1}

Simplificar

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Derivada de y=sinx/(1+lnsinx)

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