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y=e^xln(tg(x))
Derivada de la función/ tg(x)/ y=e^x/ y=e^xln(tg(x))

Derivada de y=e^xln(tg(x))

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

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Solución

Ha introducido [src] 
 x            
E *log(tan(x))
exlog⁡(tan⁡(x))e^{x} \log{\left(\tan{\left(x \right)} \right)} 
E^x*log(tan(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)=exf{\left(x \right)} = e^{x}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Derivado exe^{x} es.

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

    1. Sustituimos u=tan⁡(x)u = \tan{\left(x \right)}.

    2. Derivado log⁡(u)\log{\left(u \right)} es 1u\frac{1}{u}.

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x)\frac{d}{d x} \tan{\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)}}

      Como resultado de la secuencia de reglas:

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

    Como resultado de: (sin⁡2(x)+cos⁡2(x))excos⁡2(x)tan⁡(x)+exlog⁡(tan⁡(x))\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + e^{x} \log{\left(\tan{\left(x \right)} \right)}

  2. Simplificamos:

    (log⁡(tan⁡(x))+2sin⁡(2x))ex\left(\log{\left(\tan{\left(x \right)} \right)} + \frac{2}{\sin{\left(2 x \right)}}\right) e^{x}

Respuesta:

(log⁡(tan⁡(x))+2sin⁡(2x))ex\left(\log{\left(\tan{\left(x \right)} \right)} + \frac{2}{\sin{\left(2 x \right)}}\right) e^{x}

Gráfica
02468-8-6-4-2-1010-100000100000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                 /       2   \  x
 x               \1 + tan (x)/*e 
e *log(tan(x)) + ----------------
                      tan(x)
(tan⁡2(x)+1)extan⁡(x)+exlog⁡(tan⁡(x))\frac{\left(\tan^{2}{\left(x \right)} + 1\right) e^{x}}{\tan{\left(x \right)}} + e^{x} \log{\left(\tan{\left(x \right)} \right)}
Simplificar
Segunda derivada [src] 
/                             2                                \   
|                /       2   \      /       2   \              |   
|         2      \1 + tan (x)/    2*\1 + tan (x)/              |  x
|2 + 2*tan (x) - -------------- + --------------- + log(tan(x))|*e 
|                      2               tan(x)                  |   
\                   tan (x)                                    /
(−(tan⁡2(x)+1)2tan⁡2(x)+2(tan⁡2(x)+1)tan⁡(x)+log⁡(tan⁡(x))+2tan⁡2(x)+2)ex\left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} + 2 \tan^{2}{\left(x \right)} + 2\right) e^{x}
Simplificar
Tercera derivada [src] 
/                               2                   /                        2                  \                                \   
|                  /       2   \                    |           /       2   \      /       2   \|     /       2   \              |   
|         2      3*\1 + tan (x)/      /       2   \ |           \1 + tan (x)/    2*\1 + tan (x)/|   3*\1 + tan (x)/              |  x
|6 + 6*tan (x) - ---------------- + 2*\1 + tan (x)/*|2*tan(x) + -------------- - ---------------| + --------------- + log(tan(x))|*e 
|                       2                           |                 3               tan(x)    |        tan(x)                  |   
\                    tan (x)                        \              tan (x)                      /                                /
(−3(tan⁡2(x)+1)2tan⁡2(x)+2(tan⁡2(x)+1)((tan⁡2(x)+1)2tan⁡3(x)−2(tan⁡2(x)+1)tan⁡(x)+2tan⁡(x))+3(tan⁡2(x)+1)tan⁡(x)+log⁡(tan⁡(x))+6tan⁡2(x)+6)ex\left(- \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 2 \tan{\left(x \right)}\right) + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} + 6 \tan^{2}{\left(x \right)} + 6\right) e^{x}
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Gráfico
Derivada de y=e^xln(tg(x))
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