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Expresiones semejantes
x*exp(x)ln(x)^(sin(x))
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Derivada de la función/ x*exp/ ln(x)/ sin(x)/ exp(-x)/ x*exp(-x)ln(x)^(sin(x))

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

Solución

Ha introducido [src]

   -x    sin(x)   
x*e  *log      (x)

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

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

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x \log{\left(x \right)}^{\sin{\left(x \right)}}$ 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 $1$
  $g{\left(x \right)} = \log{\left(x \right)}^{\sin{\left(x \right)}}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$
  Como resultado de: $x \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)} + \log{\left(x \right)}^{\sin{\left(x \right)}}$
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:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Expresiones semejantes

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

Gráfico:

Definida a trozos:

Solución