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x*exp(-x)(ctg(3x))^tg(x)
Derivada de la función/ tg(x)/ tg(3x)/ x*exp/ exp(-x)/ x*exp(-x)(ctg(3x))^tg(x)

Derivada de x*exp(-x)(ctg(3x))^tg(x)

Función f() - derivada -er orden en el punto
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
   -x    tan(x)     
x*e  *cot      (3*x)
xe−xcot⁡tan⁡(x)(3x)x e^{- x} \cot^{\tan{\left(x \right)}}{\left(3 x \right)} 
(x*exp(-x))*cot(3*x)^tan(x)
Solución detallada

  1. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=xcot⁡tan⁡(x)(3x)f{\left(x \right)} = x \cot^{\tan{\left(x \right)}}{\left(3 x \right)} y g(x)=exg{\left(x \right)} = e^{x}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: xx tenemos 11

      g(x)=cot⁡tan⁡(x)(3x)g{\left(x \right)} = \cot^{\tan{\left(x \right)}}{\left(3 x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. No logro encontrar los pasos en la búsqueda de esta derivada.

        Perola derivada

        (log⁡(tan⁡(x))+1)tan⁡tan⁡(x)(x)\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)}

      Como resultado de: x(log⁡(tan⁡(x))+1)tan⁡tan⁡(x)(x)+cot⁡tan⁡(x)(3x)x \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)} + \cot^{\tan{\left(x \right)}}{\left(3 x \right)}

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Derivado exe^{x} es.

    Ahora aplicamos la regla de la derivada de una divesión:

    (−xexcot⁡tan⁡(x)(3x)+(x(log⁡(tan⁡(x))+1)tan⁡tan⁡(x)(x)+cot⁡tan⁡(x)(3x))ex)e−2x\left(- x e^{x} \cot^{\tan{\left(x \right)}}{\left(3 x \right)} + \left(x \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)} + \cot^{\tan{\left(x \right)}}{\left(3 x \right)}\right) e^{x}\right) e^{- 2 x}

  2. Simplificamos:

    (x(log⁡(tan⁡(x))+1)tan⁡tan⁡(x)(x)−xcot⁡tan⁡(x)(3x)+cot⁡tan⁡(x)(3x))e−x\left(x \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)} - x \cot^{\tan{\left(x \right)}}{\left(3 x \right)} + \cot^{\tan{\left(x \right)}}{\left(3 x \right)}\right) e^{- x}

Respuesta:

(x(log⁡(tan⁡(x))+1)tan⁡tan⁡(x)(x)−xcot⁡tan⁡(x)(3x)+cot⁡tan⁡(x)(3x))e−x\left(x \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)} - x \cot^{\tan{\left(x \right)}}{\left(3 x \right)} + \cot^{\tan{\left(x \right)}}{\left(3 x \right)}\right) e^{- x}

Gráfica
02468-8-6-4-2-1010-1000000010000000
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Primera derivada [src] 
                                                  /                              /          2     \       \    
   tan(x)      /     -x    -x\        tan(x)      |/       2   \                 \-3 - 3*cot (3*x)/*tan(x)|  -x
cot      (3*x)*\- x*e   + e  / + x*cot      (3*x)*|\1 + tan (x)/*log(cot(3*x)) + -------------------------|*e  
                                                  \                                       cot(3*x)        /
x((tan⁡2(x)+1)log⁡(cot⁡(3x))+(−3cot⁡2(3x)−3)tan⁡(x)cot⁡(3x))e−xcot⁡tan⁡(x)(3x)+(−xe−x+e−x)cot⁡tan⁡(x)(3x)x \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} + \frac{\left(- 3 \cot^{2}{\left(3 x \right)} - 3\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}}\right) e^{- x} \cot^{\tan{\left(x \right)}}{\left(3 x \right)} + \left(- x e^{- x} + e^{- x}\right) \cot^{\tan{\left(x \right)}}{\left(3 x \right)}
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Segunda derivada [src] 
               /           /                                                        2                                                2                                                                                \                                                                      \    
               |           |/                                /       2     \       \                                  /       2     \             /       2     \ /       2   \                                       |              /                                /       2     \       \|    
   tan(x)      |           ||/       2   \                 3*\1 + cot (3*x)/*tan(x)|       /       2     \          9*\1 + cot (3*x)/ *tan(x)   6*\1 + cot (3*x)/*\1 + tan (x)/     /       2   \                     |              |/       2   \                 3*\1 + cot (3*x)/*tan(x)||  -x
cot      (3*x)*|-2 + x + x*||\1 + tan (x)/*log(cot(3*x)) - ------------------------|  + 18*\1 + cot (3*x)/*tan(x) - ------------------------- - ------------------------------- + 2*\1 + tan (x)/*log(cot(3*x))*tan(x)| - 2*(-1 + x)*|\1 + tan (x)/*log(cot(3*x)) - ------------------------||*e  
               |           |\                                      cot(3*x)        /                                           2                            cot(3*x)                                                  |              \                                      cot(3*x)        /|    
               \           \                                                                                                cot (3*x)                                                                                 /                                                                      /
(x(((tan⁡2(x)+1)log⁡(cot⁡(3x))−3(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x))2−6(tan⁡2(x)+1)(cot⁡2(3x)+1)cot⁡(3x)+2(tan⁡2(x)+1)log⁡(cot⁡(3x))tan⁡(x)−9(cot⁡2(3x)+1)2tan⁡(x)cot⁡2(3x)+18(cot⁡2(3x)+1)tan⁡(x))+x−2(x−1)((tan⁡2(x)+1)log⁡(cot⁡(3x))−3(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x))−2)e−xcot⁡tan⁡(x)(3x)\left(x \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} - \frac{3 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}}\right)^{2} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(3 x \right)} + 1\right)}{\cot{\left(3 x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} \tan{\left(x \right)} - \frac{9 \left(\cot^{2}{\left(3 x \right)} + 1\right)^{2} \tan{\left(x \right)}}{\cot^{2}{\left(3 x \right)}} + 18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}\right) + x - 2 \left(x - 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} - \frac{3 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}}\right) - 2\right) e^{- x} \cot^{\tan{\left(x \right)}}{\left(3 x \right)}
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Tercera derivada [src] 
               /          /                                                        3                                                              /                                                                                                                        2       \                                                                                                                               3                            2                                                                            2                                                 \              /                                                        2                                                2                                                                                \                                                                      \    
               |          |/                                /       2     \       \      /                                /       2     \       \ |                                                                       /       2     \ /       2   \     /       2     \        |                  2                                                                                             /       2     \              /       2     \  /       2   \                                               /       2     \              /       2     \ /       2   \       |              |/                                /       2     \       \                                  /       2     \             /       2     \ /       2   \                                       |              /                                /       2     \       \|    
   tan(x)      |          ||/       2   \                 3*\1 + cot (3*x)/*tan(x)|      |/       2   \                 3*\1 + cot (3*x)/*tan(x)| |     /       2     \            /       2   \                        6*\1 + cot (3*x)/*\1 + tan (x)/   9*\1 + cot (3*x)/ *tan(x)|     /       2   \                     /       2     \ /       2   \       /       2     \                   54*\1 + cot (3*x)/ *tan(x)   27*\1 + cot (3*x)/ *\1 + tan (x)/        2    /       2   \                 108*\1 + cot (3*x)/ *tan(x)   18*\1 + cot (3*x)/*\1 + tan (x)/*tan(x)|              ||/       2   \                 3*\1 + cot (3*x)/*tan(x)|       /       2     \          9*\1 + cot (3*x)/ *tan(x)   6*\1 + cot (3*x)/*\1 + tan (x)/     /       2   \                     |              |/       2   \                 3*\1 + cot (3*x)/*tan(x)||  -x
cot      (3*x)*|3 - x + x*||\1 + tan (x)/*log(cot(3*x)) - ------------------------|  - 3*|\1 + tan (x)/*log(cot(3*x)) - ------------------------|*|- 18*\1 + cot (3*x)/*tan(x) - 2*\1 + tan (x)/*log(cot(3*x))*tan(x) + ------------------------------- + -------------------------| + 2*\1 + tan (x)/ *log(cot(3*x)) + 54*\1 + cot (3*x)/*\1 + tan (x)/ - 108*\1 + cot (3*x)/*cot(3*x)*tan(x) - -------------------------- - --------------------------------- + 4*tan (x)*\1 + tan (x)/*log(cot(3*x)) + --------------------------- - ---------------------------------------| - 3*(-1 + x)*||\1 + tan (x)/*log(cot(3*x)) - ------------------------|  + 18*\1 + cot (3*x)/*tan(x) - ------------------------- - ------------------------------- + 2*\1 + tan (x)/*log(cot(3*x))*tan(x)| + 3*(-2 + x)*|\1 + tan (x)/*log(cot(3*x)) - ------------------------||*e  
               |          |\                                      cot(3*x)        /      \                                      cot(3*x)        / |                                                                                 cot(3*x)                         2             |                                                                                                                        3                                2                                                                      cot(3*x)                            cot(3*x)               |              |\                                      cot(3*x)        /                                           2                            cot(3*x)                                                  |              \                                      cot(3*x)        /|    
               \          \                                                                                                                       \                                                                                                               cot (3*x)        /                                                                                                                     cot (3*x)                        cot (3*x)                                                                                                                            /              \                                                                                                cot (3*x)                                                                                 /                                                                      /
(x(((tan⁡2(x)+1)log⁡(cot⁡(3x))−3(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x))3−3((tan⁡2(x)+1)log⁡(cot⁡(3x))−3(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x))(6(tan⁡2(x)+1)(cot⁡2(3x)+1)cot⁡(3x)−2(tan⁡2(x)+1)log⁡(cot⁡(3x))tan⁡(x)+9(cot⁡2(3x)+1)2tan⁡(x)cot⁡2(3x)−18(cot⁡2(3x)+1)tan⁡(x))+2(tan⁡2(x)+1)2log⁡(cot⁡(3x))−27(tan⁡2(x)+1)(cot⁡2(3x)+1)2cot⁡2(3x)−18(tan⁡2(x)+1)(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x)+54(tan⁡2(x)+1)(cot⁡2(3x)+1)+4(tan⁡2(x)+1)log⁡(cot⁡(3x))tan⁡2(x)−54(cot⁡2(3x)+1)3tan⁡(x)cot⁡3(3x)+108(cot⁡2(3x)+1)2tan⁡(x)cot⁡(3x)−108(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x))−x+3(x−2)((tan⁡2(x)+1)log⁡(cot⁡(3x))−3(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x))−3(x−1)(((tan⁡2(x)+1)log⁡(cot⁡(3x))−3(cot⁡2(3x)+1)tan⁡(x)cot⁡(3x))2−6(tan⁡2(x)+1)(cot⁡2(3x)+1)cot⁡(3x)+2(tan⁡2(x)+1)log⁡(cot⁡(3x))tan⁡(x)−9(cot⁡2(3x)+1)2tan⁡(x)cot⁡2(3x)+18(cot⁡2(3x)+1)tan⁡(x))+3)e−xcot⁡tan⁡(x)(3x)\left(x \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} - \frac{3 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}}\right)^{3} - 3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} - \frac{3 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}}\right) \left(\frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(3 x \right)} + 1\right)}{\cot{\left(3 x \right)}} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} \tan{\left(x \right)} + \frac{9 \left(\cot^{2}{\left(3 x \right)} + 1\right)^{2} \tan{\left(x \right)}}{\cot^{2}{\left(3 x \right)}} - 18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}\right) + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\cot{\left(3 x \right)} \right)} - \frac{27 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(3 x \right)} + 1\right)^{2}}{\cot^{2}{\left(3 x \right)}} - \frac{18 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}} + 54 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(3 x \right)} + 1\right) + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} \tan^{2}{\left(x \right)} - \frac{54 \left(\cot^{2}{\left(3 x \right)} + 1\right)^{3} \tan{\left(x \right)}}{\cot^{3}{\left(3 x \right)}} + \frac{108 \left(\cot^{2}{\left(3 x \right)} + 1\right)^{2} \tan{\left(x \right)}}{\cot{\left(3 x \right)}} - 108 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(3 x \right)}\right) - x + 3 \left(x - 2\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} - \frac{3 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}}\right) - 3 \left(x - 1\right) \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} - \frac{3 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}}{\cot{\left(3 x \right)}}\right)^{2} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(3 x \right)} + 1\right)}{\cot{\left(3 x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\cot{\left(3 x \right)} \right)} \tan{\left(x \right)} - \frac{9 \left(\cot^{2}{\left(3 x \right)} + 1\right)^{2} \tan{\left(x \right)}}{\cot^{2}{\left(3 x \right)}} + 18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \tan{\left(x \right)}\right) + 3\right) e^{- x} \cot^{\tan{\left(x \right)}}{\left(3 x \right)}
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Gráfico
Derivada de x*exp(-x)(ctg(3x))^tg(x)
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