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y=sine^(x)/e^(x)
Derivada de la función/ e^(x)/ y=sin/ y=sine^(x)/e^(x)

Derivada de y=sine^(x)/e^(x)

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

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Solución

Ha introducido [src] 
   x   
sin (E)
-------
    x  
   E
sin⁡x(e)ex\frac{\sin^{x}{\left(e \right)}}{e^{x}} 
sin(E)^x/E^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)=sin⁡x(e)f{\left(x \right)} = \sin^{x}{\left(e \right)} y g(x)=exg{\left(x \right)} = e^{x}.

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

    1. ddxsin⁡x(e)=log⁡(sin⁡(e))sin⁡x(e)\frac{d}{d x} \sin^{x}{\left(e \right)} = \log{\left(\sin{\left(e \right)} \right)} \sin^{x}{\left(e \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:

    (−exsin⁡x(e)+exlog⁡(sin⁡(e))sin⁡x(e))e−2x\left(- e^{x} \sin^{x}{\left(e \right)} + e^{x} \log{\left(\sin{\left(e \right)} \right)} \sin^{x}{\left(e \right)}\right) e^{- 2 x}

  2. Simplificamos:

    (sin⁡(e)e)x(−1+log⁡(sin⁡(e)))\left(\frac{\sin{\left(e \right)}}{e}\right)^{x} \left(-1 + \log{\left(\sin{\left(e \right)} \right)}\right)

Respuesta:

(sin⁡(e)e)x(−1+log⁡(sin⁡(e)))\left(\frac{\sin{\left(e \right)}}{e}\right)^{x} \left(-1 + \log{\left(\sin{\left(e \right)} \right)}\right)

Gráfica
02468-8-6-4-2-1010-500000000500000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
     x     -x      x     -x            
- sin (E)*e   + sin (E)*e  *log(sin(E))
−e−xsin⁡x(e)+e−xlog⁡(sin⁡(e))sin⁡x(e)- e^{- x} \sin^{x}{\left(e \right)} + e^{- x} \log{\left(\sin{\left(e \right)} \right)} \sin^{x}{\left(e \right)}
Simplificar
Segunda derivada [src] 
   x    /       2                        \  -x
sin (E)*\1 + log (sin(E)) - 2*log(sin(E))/*e
(log⁡(sin⁡(e))2+1−2log⁡(sin⁡(e)))e−xsin⁡x(e)\left(\log{\left(\sin{\left(e \right)} \right)}^{2} + 1 - 2 \log{\left(\sin{\left(e \right)} \right)}\right) e^{- x} \sin^{x}{\left(e \right)}
Simplificar
Tercera derivada [src] 
   x    /        3                2                        \  -x
sin (E)*\-1 + log (sin(E)) - 3*log (sin(E)) + 3*log(sin(E))/*e
(3log⁡(sin⁡(e))−3log⁡(sin⁡(e))2−1+log⁡(sin⁡(e))3)e−xsin⁡x(e)\left(3 \log{\left(\sin{\left(e \right)} \right)} - 3 \log{\left(\sin{\left(e \right)} \right)}^{2} - 1 + \log{\left(\sin{\left(e \right)} \right)}^{3}\right) e^{- x} \sin^{x}{\left(e \right)}
Simplificar
Gráfico
Derivada de y=sine^(x)/e^(x)
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