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y=cos^2x+lntgx/2x*exp(-x)
Derivada de la función/ cos^2/ lntgx/ tgx/2/ x+lnt/ y=cos/ x*exp/ exp(-x)/ y=cos^2x+lntgx/2/ y=cos^2x+lntgx/2x*exp(-x)

Derivada de y=cos^2x+lntgx/2x*exp(-x)

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

Ha introducido [src] 
   2      log(tan(x))    -x
cos (x) + -----------*x*e  
               2
xlog⁡(tan⁡(x))2e−x+cos⁡2(x)x \frac{\log{\left(\tan{\left(x \right)} \right)}}{2} e^{- x} + \cos^{2}{\left(x \right)} 
cos(x)^2 + ((log(tan(x))/2)*x)*exp(-x)
Solución detallada

  1. diferenciamos xlog⁡(tan⁡(x))2e−x+cos⁡2(x)x \frac{\log{\left(\tan{\left(x \right)} \right)}}{2} e^{- x} + \cos^{2}{\left(x \right)} miembro por miembro:

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

    2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

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

      Como resultado de la secuencia de reglas:

      −2sin⁡(x)cos⁡(x)- 2 \sin{\left(x \right)} \cos{\left(x \right)}

    4. 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)=xlog⁡(tan⁡(x))f{\left(x \right)} = x \log{\left(\tan{\left(x \right)} \right)} y g(x)=2exg{\left(x \right)} = 2 e^{x}.

      Para calcular ddxf(x)\frac{d}{d x} f{\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)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Según el principio, aplicamos: xx tenemos 11

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

      Para calcular ddxg(x)\frac{d}{d x} g{\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. Derivado exe^{x} es.

        Entonces, como resultado: 2ex2 e^{x}

      Ahora aplicamos la regla de la derivada de una divesión:

      (−2xexlog⁡(tan⁡(x))+2(x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡(x)+log⁡(tan⁡(x)))ex)e−2x4\frac{\left(- 2 x e^{x} \log{\left(\tan{\left(x \right)} \right)} + 2 \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)}\right) e^{x}\right) e^{- 2 x}}{4}

    Como resultado de: (−2xexlog⁡(tan⁡(x))+2(x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡(x)+log⁡(tan⁡(x)))ex)e−2x4−2sin⁡(x)cos⁡(x)\frac{\left(- 2 x e^{x} \log{\left(\tan{\left(x \right)} \right)} + 2 \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)}\right) e^{x}\right) e^{- 2 x}}{4} - 2 \sin{\left(x \right)} \cos{\left(x \right)}

  2. Simplificamos:

    (−xlog⁡(tan⁡(x))+xsin⁡(x)cos⁡(x)+4exsin⁡3(x)cos⁡(x)−4extan⁡(x)+log⁡(tan⁡(x)))e−x2\frac{\left(- x \log{\left(\tan{\left(x \right)} \right)} + \frac{x}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{4 e^{x} \sin^{3}{\left(x \right)}}{\cos{\left(x \right)}} - 4 e^{x} \tan{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)}\right) e^{- x}}{2}

Respuesta:

(−xlog⁡(tan⁡(x))+xsin⁡(x)cos⁡(x)+4exsin⁡3(x)cos⁡(x)−4extan⁡(x)+log⁡(tan⁡(x)))e−x2\frac{\left(- x \log{\left(\tan{\left(x \right)} \right)} + \frac{x}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{4 e^{x} \sin^{3}{\left(x \right)}}{\cos{\left(x \right)}} - 4 e^{x} \tan{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)}\right) e^{- x}}{2}

Gráfica
02468-8-6-4-2-1010-25000002500000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
/                /       2   \\                            -x            
|log(tan(x))   x*\1 + tan (x)/|  -x                     x*e  *log(tan(x))
|----------- + ---------------|*e   - 2*cos(x)*sin(x) - -----------------
\     2            2*tan(x)   /                                 2
−xe−xlog⁡(tan⁡(x))2+(x(tan⁡2(x)+1)2tan⁡(x)+log⁡(tan⁡(x))2)e−x−2sin⁡(x)cos⁡(x)- \frac{x e^{- x} \log{\left(\tan{\left(x \right)} \right)}}{2} + \left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{2 \tan{\left(x \right)}} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{2}\right) e^{- x} - 2 \sin{\left(x \right)} \cos{\left(x \right)}
Simplificar
Segunda derivada [src] 
                                                                                                                    /                 /       2   \\                          
                          /  /       2   \              \                                             /       2   \ |        2      x*\1 + tan (x)/|  -x                      
                          |x*\1 + tan (x)/              |  -x                                         \1 + tan (x)/*|2*x + ------ - ---------------|*e                        
                          |--------------- + log(tan(x))|*e      -x                  -x                             |      tan(x)          2       |         /       2   \  -x
       2           2      \     tan(x)                  /       e  *log(tan(x))   x*e  *log(tan(x))                 \                   tan (x)    /       x*\1 + tan (x)/*e  
- 2*cos (x) + 2*sin (x) - ----------------------------------- - --------------- + ----------------- + -------------------------------------------------- - -------------------
                                           2                           2                  2                                   2                                  2*tan(x)
−x(tan⁡2(x)+1)e−x2tan⁡(x)+xe−xlog⁡(tan⁡(x))2−(x(tan⁡2(x)+1)tan⁡(x)+log⁡(tan⁡(x)))e−x2+(tan⁡2(x)+1)(−x(tan⁡2(x)+1)tan⁡2(x)+2x+2tan⁡(x))e−x2+2sin⁡2(x)−2cos⁡2(x)−e−xlog⁡(tan⁡(x))2- \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x}}{2 \tan{\left(x \right)}} + \frac{x e^{- x} \log{\left(\tan{\left(x \right)} \right)}}{2} - \frac{\left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)}\right) e^{- x}}{2} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + 2 x + \frac{2}{\tan{\left(x \right)}}\right) e^{- x}}{2} + 2 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - \frac{e^{- x} \log{\left(\tan{\left(x \right)} \right)}}{2}
Simplificar
Tercera derivada [src] 
                                                        /                               2                    2                    3                           \                                                                                                                                                                                      
                                                        |                  /       2   \        /       2   \        /       2   \                            |                                                                                                                                                                                      
                  /  /       2   \              \       |         2      3*\1 + tan (x)/    4*x*\1 + tan (x)/    2*x*\1 + tan (x)/        /       2   \       |  -x                                                                                                                                                                                  
                  |x*\1 + tan (x)/              |  -x   |6 + 6*tan (x) - ---------------- - ------------------ + ------------------ + 4*x*\1 + tan (x)/*tan(x)|*e                                                                                                                                                                               2    
                  |--------------- + log(tan(x))|*e     |                       2                 tan(x)                 3                                    |                                               /       2   \  -x                 /                 /       2   \\          -x                 /       2   \  -x     /       2   \   -x
 -x               \     tan(x)                  /       \                    tan (x)                                  tan (x)                                 /                           /       2   \  -x   \1 + tan (x)/*e     /       2   \ |        2      x*\1 + tan (x)/|  -x   x*e  *log(tan(x))   x*\1 + tan (x)/*e     x*\1 + tan (x)/ *e  
e  *log(tan(x)) + ----------------------------------- + ----------------------------------------------------------------------------------------------------------- + 8*cos(x)*sin(x) - x*\1 + tan (x)/*e   - ----------------- - \1 + tan (x)/*|2*x + ------ - ---------------|*e   - ----------------- + ------------------- + --------------------
                                   2                                                                         2                                                                                                      tan(x)                      |      tan(x)          2       |               2                  tan(x)                   2         
                                                                                                                                                                                                                                                \                   tan (x)    /                                                      2*tan (x)
x(tan⁡2(x)+1)2e−x2tan⁡2(x)−x(tan⁡2(x)+1)e−x+x(tan⁡2(x)+1)e−xtan⁡(x)−xe−xlog⁡(tan⁡(x))2+(x(tan⁡2(x)+1)tan⁡(x)+log⁡(tan⁡(x)))e−x2−(tan⁡2(x)+1)(−x(tan⁡2(x)+1)tan⁡2(x)+2x+2tan⁡(x))e−x−(tan⁡2(x)+1)e−xtan⁡(x)+(2x(tan⁡2(x)+1)3tan⁡3(x)−4x(tan⁡2(x)+1)2tan⁡(x)+4x(tan⁡2(x)+1)tan⁡(x)−3(tan⁡2(x)+1)2tan⁡2(x)+6tan⁡2(x)+6)e−x2+8sin⁡(x)cos⁡(x)+e−xlog⁡(tan⁡(x))\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{- x}}{2 \tan^{2}{\left(x \right)}} - x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} + \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x}}{\tan{\left(x \right)}} - \frac{x e^{- x} \log{\left(\tan{\left(x \right)} \right)}}{2} + \frac{\left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)}\right) e^{- x}}{2} - \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + 2 x + \frac{2}{\tan{\left(x \right)}}\right) e^{- x} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) e^{- x}}{\tan{\left(x \right)}} + \frac{\left(\frac{2 x \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{3}{\left(x \right)}} - \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan{\left(x \right)}} + 4 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 6 \tan^{2}{\left(x \right)} + 6\right) e^{- x}}{2} + 8 \sin{\left(x \right)} \cos{\left(x \right)} + e^{- x} \log{\left(\tan{\left(x \right)} \right)}
Simplificar
Gráfico
Derivada de y=cos^2x+lntgx/2x*exp(-x)
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