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п^2/(arctge^x)x*exp(-x)
Derivada de la función/ arctg/ x*exp/ exp(-x)/ п^2/(arctge^x)x*exp(-x)

Derivada de п^2/(arctge^x)x*exp(-x)

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src] 
    2         
  pi        -x
--------*x*e  
    x         
atan (E)
xπ2atan⁡x(e)e−xx \frac{\pi^{2}}{\operatorname{atan}^{x}{\left(e \right)}} e^{- x} 
((pi^2/atan(E)^x)*x)*exp(-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)=π2xf{\left(x \right)} = \pi^{2} x y g(x)=exatan⁡x(e)g{\left(x \right)} = e^{x} \operatorname{atan}^{x}{\left(e \right)}.

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

    1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

      1. Según el principio, aplicamos: xx tenemos 11

      Entonces, como resultado: π2\pi^{2}

    Para calcular ddxg(x)\frac{d}{d x} g{\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)=atan⁡x(e)f{\left(x \right)} = \operatorname{atan}^{x}{\left(e \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. ddxatan⁡x(e)=log⁡(atan⁡(e))atan⁡x(e)\frac{d}{d x} \operatorname{atan}^{x}{\left(e \right)} = \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)}

      g(x)=exg{\left(x \right)} = e^{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Derivado exe^{x} es.

      Como resultado de: exlog⁡(atan⁡(e))atan⁡x(e)+exatan⁡x(e)e^{x} \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)} + e^{x} \operatorname{atan}^{x}{\left(e \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    (−π2x(exlog⁡(atan⁡(e))atan⁡x(e)+exatan⁡x(e))+π2exatan⁡x(e))e−2xatan⁡−2x(e)\left(- \pi^{2} x \left(e^{x} \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)} + e^{x} \operatorname{atan}^{x}{\left(e \right)}\right) + \pi^{2} e^{x} \operatorname{atan}^{x}{\left(e \right)}\right) e^{- 2 x} \operatorname{atan}^{- 2 x}{\left(e \right)}

  2. Simplificamos:

    π2(1eatan⁡(e))x(−x−xlog⁡(atan⁡(e))+1)\pi^{2} \left(\frac{1}{e \operatorname{atan}{\left(e \right)}}\right)^{x} \left(- x - x \log{\left(\operatorname{atan}{\left(e \right)} \right)} + 1\right)

Respuesta:

π2(1eatan⁡(e))x(−x−xlog⁡(atan⁡(e))+1)\pi^{2} \left(\frac{1}{e \operatorname{atan}{\left(e \right)}}\right)^{x} \left(- x - x \log{\left(\operatorname{atan}{\left(e \right)} \right)} + 1\right)

Gráfica
02468-8-6-4-2-1010-5000000050000000
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Primera derivada [src] 
/    2                                  \                          
|  pi           2     -x                |  -x       2     -x     -x
|-------- - x*pi *atan  (E)*log(atan(E))|*e   - x*pi *atan  (E)*e  
|    x                                  |                          
\atan (E)                               /
−π2xe−xatan⁡−x(e)+(−π2xlog⁡(atan⁡(e))atan⁡−x(e)+π2atan⁡x(e))e−x- \pi^{2} x e^{- x} \operatorname{atan}^{- x}{\left(e \right)} + \left(- \pi^{2} x \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{- x}{\left(e \right)} + \frac{\pi^{2}}{\operatorname{atan}^{x}{\left(e \right)}}\right) e^{- x}
Simplificar
Segunda derivada [src] 
  2     -x                                                                      -x
pi *atan  (E)*(-2 + x + (-2 + x*log(atan(E)))*log(atan(E)) + 2*x*log(atan(E)))*e
π2(2xlog⁡(atan⁡(e))+x+(xlog⁡(atan⁡(e))−2)log⁡(atan⁡(e))−2)e−xatan⁡−x(e)\pi^{2} \left(2 x \log{\left(\operatorname{atan}{\left(e \right)} \right)} + x + \left(x \log{\left(\operatorname{atan}{\left(e \right)} \right)} - 2\right) \log{\left(\operatorname{atan}{\left(e \right)} \right)} - 2\right) e^{- x} \operatorname{atan}^{- x}{\left(e \right)}
Simplificar
Tercera derivada [src] 
   2     -x    /            2                                                                                         \  -x
-pi *atan  (E)*\-3 + x + log (atan(E))*(-3 + x*log(atan(E))) + 3*x*log(atan(E)) + 3*(-2 + x*log(atan(E)))*log(atan(E))/*e
−π2(3xlog⁡(atan⁡(e))+x+(xlog⁡(atan⁡(e))−3)log⁡(atan⁡(e))2+3(xlog⁡(atan⁡(e))−2)log⁡(atan⁡(e))−3)e−xatan⁡−x(e)- \pi^{2} \left(3 x \log{\left(\operatorname{atan}{\left(e \right)} \right)} + x + \left(x \log{\left(\operatorname{atan}{\left(e \right)} \right)} - 3\right) \log{\left(\operatorname{atan}{\left(e \right)} \right)}^{2} + 3 \left(x \log{\left(\operatorname{atan}{\left(e \right)} \right)} - 2\right) \log{\left(\operatorname{atan}{\left(e \right)} \right)} - 3\right) e^{- x} \operatorname{atan}^{- x}{\left(e \right)}
Simplificar
Gráfico
Derivada de п^2/(arctge^x)x*exp(-x)
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