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tgx*e^x
Derivada de la función/ x*e^x/ tgx*e^x

Derivada de tgx*e^x

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

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Definida a trozos:

Solución

Ha introducido [src] 
        x
tan(x)*E
extan⁡(x)e^{x} \tan{\left(x \right)} 
tan(x)*E^x
Solución detallada

  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)=tan⁡(x)f{\left(x \right)} = \tan{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Reescribimos las funciones para diferenciar:

      tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

    2. 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)f{\left(x \right)} = \sin{\left(x \right)} y g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

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

      1. La derivada del seno es igual al coseno:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

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

      1. La derivada del coseno es igual a menos el seno:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      Ahora aplicamos la regla de la derivada de una divesión:

      sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \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: (sin⁡2(x)+cos⁡2(x))excos⁡2(x)+extan⁡(x)\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x}}{\cos^{2}{\left(x \right)}} + e^{x} \tan{\left(x \right)}

  2. Simplificamos:

    (sin⁡(2x)2+1)excos⁡2(x)\frac{\left(\frac{\sin{\left(2 x \right)}}{2} + 1\right) e^{x}}{\cos^{2}{\left(x \right)}}

Respuesta:

(sin⁡(2x)2+1)excos⁡2(x)\frac{\left(\frac{\sin{\left(2 x \right)}}{2} + 1\right) e^{x}}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-500000500000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
/       2   \  x    x       
\1 + tan (x)/*e  + e *tan(x)
(tan⁡2(x)+1)ex+extan⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) e^{x} + e^{x} \tan{\left(x \right)}
Simplificar
Segunda derivada [src] 
/         2        /       2   \                \  x
\2 + 2*tan (x) + 2*\1 + tan (x)/*tan(x) + tan(x)/*e
(2(tan⁡2(x)+1)tan⁡(x)+2tan⁡2(x)+tan⁡(x)+2)ex\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) e^{x}
Simplificar
Tercera derivada [src] 
/         2        /       2   \ /         2   \     /       2   \                \  x
\3 + 3*tan (x) + 2*\1 + tan (x)/*\1 + 3*tan (x)/ + 6*\1 + tan (x)/*tan(x) + tan(x)/*e
(2(tan⁡2(x)+1)(3tan⁡2(x)+1)+6(tan⁡2(x)+1)tan⁡(x)+3tan⁡2(x)+tan⁡(x)+3)ex\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 3\right) e^{x}
Simplificar
Gráfico
Derivada de tgx*e^x
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