Sr Examen

Otras calculadoras

y=cos*log*tg*e^sinx
Derivada de la función/ y=cos/ e^sinx/ y=cos*log*tg*e^sinx

Derivada de y=cos*log*tg*e^sinx

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
                    sin(x)
cos(E)*log(tan(x))*E
esin⁡(x)log⁡(tan⁡(x))cos⁡(e)e^{\sin{\left(x \right)}} \log{\left(\tan{\left(x \right)} \right)} \cos{\left(e \right)} 
(cos(E)*log(tan(x)))*E^sin(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)=log⁡(tan⁡(x))cos⁡(e)f{\left(x \right)} = \log{\left(\tan{\left(x \right)} \right)} \cos{\left(e \right)}; calculamos 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. 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)}}

      Entonces, como resultado: (sin⁡2(x)+cos⁡2(x))cos⁡(e)cos⁡2(x)tan⁡(x)\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cos{\left(e \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}

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

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

    2. Derivado eue^{u} es.

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

      Como resultado de la secuencia de reglas:

      esin⁡(x)cos⁡(x)e^{\sin{\left(x \right)}} \cos{\left(x \right)}

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

  2. Simplificamos:

    (−log⁡(tan⁡(x))sin⁡2(x)+log⁡(tan⁡(x))+1sin⁡(x))esin⁡(x)cos⁡(e)cos⁡(x)\frac{\left(- \log{\left(\tan{\left(x \right)} \right)} \sin^{2}{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)} + \frac{1}{\sin{\left(x \right)}}\right) e^{\sin{\left(x \right)}} \cos{\left(e \right)}}{\cos{\left(x \right)}}

Respuesta:

(−log⁡(tan⁡(x))sin⁡2(x)+log⁡(tan⁡(x))+1sin⁡(x))esin⁡(x)cos⁡(e)cos⁡(x)\frac{\left(- \log{\left(\tan{\left(x \right)} \right)} \sin^{2}{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)} + \frac{1}{\sin{\left(x \right)}}\right) e^{\sin{\left(x \right)}} \cos{\left(e \right)}}{\cos{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-100100
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
/       2   \         sin(x)                                    
\1 + tan (x)/*cos(E)*e                        sin(x)            
---------------------------- + cos(E)*cos(x)*e      *log(tan(x))
           tan(x)
(tan⁡2(x)+1)esin⁡(x)cos⁡(e)tan⁡(x)+esin⁡(x)log⁡(tan⁡(x))cos⁡(e)cos⁡(x)\frac{\left(\tan^{2}{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}} \cos{\left(e \right)}}{\tan{\left(x \right)}} + e^{\sin{\left(x \right)}} \log{\left(\tan{\left(x \right)} \right)} \cos{\left(e \right)} \cos{\left(x \right)}
Simplificar
Segunda derivada [src] 
/                             2                                                            \               
|                /       2   \                                         /       2   \       |               
|         2      \1 + tan (x)/    /     2            \               2*\1 + tan (x)/*cos(x)|         sin(x)
|2 + 2*tan (x) - -------------- - \- cos (x) + sin(x)/*log(tan(x)) + ----------------------|*cos(E)*e      
|                      2                                                     tan(x)        |               
\                   tan (x)                                                                /
(−(sin⁡(x)−cos⁡2(x))log⁡(tan⁡(x))−(tan⁡2(x)+1)2tan⁡2(x)+2(tan⁡2(x)+1)cos⁡(x)tan⁡(x)+2tan⁡2(x)+2)esin⁡(x)cos⁡(e)\left(- \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\tan{\left(x \right)} \right)} - \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) \cos{\left(x \right)}}{\tan{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right) e^{\sin{\left(x \right)}} \cos{\left(e \right)}
Simplificar
Tercera derivada [src] 
/                /                        2                  \     /                             2\                                                                                            \               
|                |           /       2   \      /       2   \|     |                /       2   \ |                                                          /       2   \ /     2            \|               
|  /       2   \ |           \1 + tan (x)/    2*\1 + tan (x)/|     |         2      \1 + tan (x)/ |          /       2              \                      3*\1 + tan (x)/*\- cos (x) + sin(x)/|         sin(x)
|2*\1 + tan (x)/*|2*tan(x) + -------------- - ---------------| + 3*|2 + 2*tan (x) - --------------|*cos(x) - \1 - cos (x) + 3*sin(x)/*cos(x)*log(tan(x)) - ------------------------------------|*cos(E)*e      
|                |                 3               tan(x)    |     |                      2       |                                                                       tan(x)               |               
\                \              tan (x)                      /     \                   tan (x)    /                                                                                            /
(−3(sin⁡(x)−cos⁡2(x))(tan⁡2(x)+1)tan⁡(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)2tan⁡2(x)+2tan⁡2(x)+2)cos⁡(x)−(3sin⁡(x)−cos⁡2(x)+1)log⁡(tan⁡(x))cos⁡(x))esin⁡(x)cos⁡(e)\left(- \frac{3 \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\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) + 3 \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right) \cos{\left(x \right)} - \left(3 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)}\right) e^{\sin{\left(x \right)}} \cos{\left(e \right)}
Simplificar
Gráfico
Derivada de y=cos*log*tg*e^sinx
    © Sr Examen – Калькулятор