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Derivada de la función/ x*exp/ exp(-x)/ sqrt(x)/ sin(x)/ x*exp(-x)(sin(x))^sqrt(x)

Derivada de x*exp(-x)(sin(x))^sqrt(x)

Solución

Ha introducido [src]

                ___
   -x         \/ x 
x*e  *(sin(x))

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

(x*exp(-x))*sin(x)^(sqrt(x))

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \sin^{\sqrt{x}}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right)$
  Como resultado de: $x x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right) + \sin^{\sqrt{x}}{\left(x \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Derivado $e^{x}$ es.
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

              /          /                              3                                                                                                                                                                                       \                                                          /                                2                                                         \\    
              |          |/                  ___       \              /                  ___       \ /                                           ___    2   \                           2                          ___    3           ___       |              /                  ___       \              |  /                  ___       \                                               ___    2   ||    
              |          ||log(sin(x))   2*\/ x *cos(x)|      12      |log(sin(x))   2*\/ x *cos(x)| |    ___   log(sin(x))     4*cos(x)     4*\/ x *cos (x)|   3*log(sin(x))     12*cos (x)      6*cos(x)    16*\/ x *cos (x)   16*\/ x *cos(x)|              |log(sin(x))   2*\/ x *cos(x)|              |  |log(sin(x))   2*\/ x *cos(x)|        ___   log(sin(x))     4*cos(x)     4*\/ x *cos (x)||    
              |        x*||----------- + --------------|  - ----- - 3*|----------- + --------------|*|4*\/ x  + ----------- - ------------ + ---------------| + ------------- - ------------- - ----------- + ---------------- + ---------------|   3*(-2 + x)*|----------- + --------------|   3*(-1 + x)*|- |----------- + --------------|  + 4*\/ x  + ----------- - ------------ + ---------------||    
          ___ |          ||     ___          sin(x)    |      ___     |     ___          sin(x)    | |               3/2        ___                 2       |         5/2         ___    2       3/2                 3                sin(x)    |              |     ___          sin(x)    |              |  |     ___          sin(x)    |                   3/2        ___                 2       ||    
        \/ x  |          \\   \/ x                     /    \/ x      \   \/ x                     / \              x         \/ x *sin(x)       sin (x)    /        x          \/ x *sin (x)   x   *sin(x)       sin (x)                       /              \   \/ x                     /              \  \   \/ x                     /                  x         \/ x *sin(x)       sin (x)    /|  -x
(sin(x))     *|3 - x + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------- + -------------------------------------------------------------------------------------------------------|*e  
              \                                                                                                                    8                                                                                                                                    2                                                                          4                                                   /

$$\left(\frac{x \left(\frac{16 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{16 \sqrt{x} \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \left(\frac{2 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\right)^{3} - 3 \left(\frac{2 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\right) \left(4 \sqrt{x} + \frac{4 \sqrt{x} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{4 \cos{\left(x \right)}}{\sqrt{x} \sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{x^{\frac{3}{2}}}\right) - \frac{12}{\sqrt{x}} - \frac{12 \cos^{2}{\left(x \right)}}{\sqrt{x} \sin^{2}{\left(x \right)}} - \frac{6 \cos{\left(x \right)}}{x^{\frac{3}{2}} \sin{\left(x \right)}} + \frac{3 \log{\left(\sin{\left(x \right)} \right)}}{x^{\frac{5}{2}}}\right)}{8} - x + \frac{3 \left(x - 2\right) \left(\frac{2 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\right)}{2} + \frac{3 \left(x - 1\right) \left(4 \sqrt{x} + \frac{4 \sqrt{x} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \left(\frac{2 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\right)^{2} - \frac{4 \cos{\left(x \right)}}{\sqrt{x} \sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{x^{\frac{3}{2}}}\right)}{4} + 3\right) e^{- x} \sin^{\sqrt{x}}{\left(x \right)}$$

Simplificar

Gráfico

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Derivada de x*exp(-x)(sin(x))^sqrt(x)

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