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Derivada de:
Derivada de 1/(x+3)
Derivada de 4*y
Derivada de (2x-7)^8
Derivada de (-1)/x-3*x
Expresiones idénticas
(е^-x):((sin^ dos)x)
(е en el grado menos x):(( seno de al cuadrado )x)
(е en el grado menos x):(( seno de en el grado dos)x)
(е-x):((sin2)x)
е-x:sin2x
(е^-x):((sin²)x)
(е en el grado -x):((sin en el grado 2)x)
е^-x:sin^2x
Expresiones semejantes
(е^+x):((sin^2)x)
Expresiones con funciones
Seno sin
sin^5*x
sin(x)^(2)*2^x^2
sin^4(x^2-7)
sin(x^2-7)^(4)
sin^n(x)

Derivada de la función/ sin^2/ (е^-x):((sin^2)x)

Derivada de (е^-x):((sin^2)x)

Solución

Ha introducido [src]

    -x   
   E     
---------
   2     
sin (x)*x

$$\frac{e^{- x}}{x \sin^{2}{\left(x \right)}}$$

E^(-x)/((sin(x)^2*x))

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. La derivada de una constante $1$ es igual a cero.
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \sin^{2}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = \sin{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de la secuencia de reglas:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  $h{\left(x \right)} = e^{x}$ ; calculamos $\frac{d}{d x} h{\left(x \right)}$ :
  1. Derivado $e^{x}$ es.
  Como resultado de: $x e^{x} \sin^{2}{\left(x \right)} + 2 x e^{x} \sin{\left(x \right)} \cos{\left(x \right)} + e^{x} \sin^{2}{\left(x \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

 /                                                                                                                                                                  /1   2*cos(x)\                                                                                                                               /1   2*cos(x)\ /     2           2                     \                                        /1   2*cos(x)\                                                                                                                                                                                                                                        \     
 |                 /       2           2                       \                           /              2              \                                          |- + --------|*(2*x*cos(x) + sin(x))      /     2           2                     \            /     2           2                     \   2*|- + --------|*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/         2                            2*|- + --------|*(2*x*cos(x) + sin(x))*cos(x)                                                                                                                                                                                                           |     
 |               2*\- 3*cos (x) + 3*sin (x) + 4*x*cos(x)*sin(x)/                           |    1    3*cos (x)   2*cos(x)|   3*(2*x*cos(x) + sin(x))                \x    sin(x) /                         12*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*cos(x)   6*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/     \x    sin(x) /                                             10*cos (x)*(2*x*cos(x) + sin(x))     \x    sin(x) /                                8*(2*x*cos(x) + sin(x))*cos(x)                               /                                                               /     2           2                     \                                 \|     
 |    2*sin(x) - ----------------------------------------------- + 2*(2*x*cos(x) + sin(x))*|1 + -- + --------- + --------| + ----------------------- + 4*x*cos(x) + ------------------------------------ - --------------------------------------------------- - ------------------------------------------- - ---------------------------------------------------------- + -------------------------------- + --------------------------------------------- + ------------------------------                               |2*x*cos(x) + sin(x)   /1   2*cos(x)\                         2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/   2*(2*x*cos(x) + sin(x))*cos(x)||     
 |                                    sin(x)                                               |     2       2       x*sin(x)|               2                                           x                                              2                                              x*sin(x)                                              sin(x)                                            2                                       sin(x)                                 x*sin(x)                                        3*|------------------- + |- + --------|*(2*x*cos(x) + sin(x)) - ------------------------------------------- + ------------------------------||     
 |                                                                                         \    x     sin (x)            /              x                                                                                        sin (x)                                                                                                                                                sin (x)                                                                                                 3*(2*x*cos(x) + sin(x))     \         x            \x    sin(x) /                                            sin(x)                                 sin(x)            /|  -x 
-|1 + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------|*e   
 \                                                                                                                                                                                                                                            x*sin(x)                                                                                                                                                                                                                                                  x*sin(x)                                                                             x*sin(x)                                                                  /     
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                       2                                                                                                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                                                                                                                                  x*sin (x)

$$- \frac{\left(1 + \frac{3 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}} + \frac{3 \left(\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right) + \frac{2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{2 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\sin{\left(x \right)}} + \frac{2 x \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right)}{x \sin{\left(x \right)}} + \frac{4 x \cos{\left(x \right)} + \frac{2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(1 + \frac{3 \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) + \frac{10 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right) \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\sin{\left(x \right)}} - \frac{12 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \left(4 x \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right)}{x} + \frac{8 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{6 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{x \sin{\left(x \right)}} + \frac{3 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x^{2}}}{x \sin{\left(x \right)}}\right) e^{- x}}{x \sin^{2}{\left(x \right)}}$$

Simplificar

Gráfico

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de (е^-x):((sin^2)x)

Gráfico:

Definida a trozos:

Solución