Derivada de y=(lnx+1)/(x^2sinx)

Ha introducido [src]

log(x) + 1
----------
 2        
x *sin(x)

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

(log(x) + 1)/((x^2*sin(x)))

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = \log{\left(x \right)} + 1$ y $g{\left(x \right)} = x^{2} \sin{\left(x \right)}$ .

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

$\frac{x \sin{\left(x \right)} - \left(x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}\right) \left(\log{\left(x \right)} + 1\right)}{x^{4} \sin^{2}{\left(x \right)}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

/    1    \                                          
|---------|                                          
| 2       |   /   2                    \             
\x *sin(x)/   \- x *cos(x) - 2*x*sin(x)/*(log(x) + 1)
----------- + ---------------------------------------
     x                        4    2                 
                             x *sin (x)

\frac{\frac{1}{x^{2}} \frac{1}{\sin{\left(x \right)}}}{x} + \frac{\left(- x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}\right) \left(\log{\left(x \right)} + 1\right)}{x^{4} \sin^{2}{\left(x \right)}}

Simplificar

Segunda derivada [src]

                   /                                                 2                                                                             \                          
                   |/2   cos(x)\                         2*sin(x) - x *sin(x) + 4*x*cos(x)   2*(2*sin(x) + x*cos(x))   (2*sin(x) + x*cos(x))*cos(x)|                          
      (1 + log(x))*||- + ------|*(2*sin(x) + x*cos(x)) - --------------------------------- + ----------------------- + ----------------------------|                          
  1                \\x   sin(x)/                                         x                              x                         sin(x)           /   2*(2*sin(x) + x*cos(x))
- - + ---------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------
  x                                                                       sin(x)                                                                               x*sin(x)       
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                   3                                                                                          
                                                                                  x *sin(x)

\frac{\frac{\left(\log{\left(x \right)} + 1\right) \left(\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x}\right)}{\sin{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}} - \frac{1}{x}}{x^{3} \sin{\left(x \right)}}

Simplificar

Tercera derivada [src]

                  /                                                                                                                                                                                      /2   cos(x)\ /            2                    \     /2   cos(x)\                                                           /2   cos(x)\                                                                                                             \                                                                                                                                                                
                  |                                            /              2              \                2                         /            2                    \                              |- + ------|*\2*sin(x) - x *sin(x) + 4*x*cos(x)/   2*|- + ------|*(2*sin(x) + x*cos(x))        2                            |- + ------|*(2*sin(x) + x*cos(x))*cos(x)     /            2                    \                                        |                                                                                                                                                                
                  |                                            |    6    2*cos (x)   4*cos(x)|   -6*cos(x) + x *cos(x) + 6*x*sin(x)   6*\2*sin(x) - x *sin(x) + 4*x*cos(x)/   10*(2*sin(x) + x*cos(x))   \x   sin(x)/                                         \x   sin(x)/                         3*cos (x)*(2*sin(x) + x*cos(x))   \x   sin(x)/                                3*\2*sin(x) - x *sin(x) + 4*x*cos(x)/*cos(x)   8*(2*sin(x) + x*cos(x))*cos(x)|     /                                                 2                                                                             \                          
     (1 + log(x))*|2*sin(x) + x*cos(x) + (2*sin(x) + x*cos(x))*|1 + -- + --------- + --------| - ---------------------------------- - ------------------------------------- + ------------------------ - ------------------------------------------------ + ------------------------------------ + ------------------------------- + ----------------------------------------- - -------------------------------------------- + ------------------------------|     |/2   cos(x)\                         2*sin(x) - x *sin(x) + 4*x*cos(x)   2*(2*sin(x) + x*cos(x))   (2*sin(x) + x*cos(x))*cos(x)|                          
                  |                                            |     2       2       x*sin(x)|                   x                                       2                                2                                     x                                            x                                    2                                    sin(x)                                      x*sin(x)                                x*sin(x)           |   3*||- + ------|*(2*sin(x) + x*cos(x)) - --------------------------------- + ----------------------- + ----------------------------|                          
2                 \                                            \    x     sin (x)            /                                                          x                                x                                                                                                                     sin (x)                                                                                                                                        /     \\x   sin(x)/                                         x                              x                         sin(x)           /   3*(2*sin(x) + x*cos(x))
-- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + -----------------------
 2                                                                                                                                                                                                                             sin(x)                                                                                                                                                                                                                                                                                           x*sin(x)                                                                        2              
x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              x *sin(x)       
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                            3                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                                                                           x *sin(x)

\frac{- \frac{\left(\log{\left(x \right)} + 1\right) \left(x \cos{\left(x \right)} + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{x \sin{\left(x \right)}} + \frac{6}{x^{2}}\right) + \frac{3 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right)}{x} + \frac{8 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{3 \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{x^{2} \cos{\left(x \right)} + 6 x \sin{\left(x \right)} - 6 \cos{\left(x \right)}}{x} + \frac{10 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2}} - \frac{6 \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2}}\right)}{\sin{\left(x \right)}} + \frac{3 \left(\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x}\right)}{x \sin{\left(x \right)}} + \frac{3 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2} \sin{\left(x \right)}} + \frac{2}{x^{2}}}{x^{3} \sin{\left(x \right)}}

Simplificar

Gráfico

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Derivada de y=(lnx+1)/(x^2sinx)

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