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y=cosx*ln(tgx)-ln(tgx/2)

Derivada de y=cosx*ln(tgx)-ln(tgx/2)

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Ha introducido [src]
                        /tan(x)\
cos(x)*log(tan(x)) - log|------|
                        \  2   /
log(tan(x)2)+log(tan(x))cos(x)- \log{\left(\frac{\tan{\left(x \right)}}{2} \right)} + \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)}
cos(x)*log(tan(x)) - log(tan(x)/2)
Solución detallada
  1. diferenciamos log(tan(x)2)+log(tan(x))cos(x)- \log{\left(\frac{\tan{\left(x \right)}}{2} \right)} + \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)} miembro por miembro:

    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)=cos(x)f{\left(x \right)} = \cos{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\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)}

      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:

          sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        sin2(x)+cos2(x)cos2(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: sin2(x)+cos2(x)cos(x)tan(x)log(tan(x))sin(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}} - \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}

    2. 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)2u = \frac{\tan{\left(x \right)}}{2}.

      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)2\frac{d}{d x} \frac{\tan{\left(x \right)}}{2}:

        1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

          1. ddxtan(x)=1cos2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

          Entonces, como resultado: sin2(x)+cos2(x)2cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        sin2(x)+cos2(x)cos2(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: sin2(x)+cos2(x)cos2(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: sin2(x)+cos2(x)cos(x)tan(x)sin2(x)+cos2(x)cos2(x)tan(x)log(tan(x))sin(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} - \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}

  2. Simplificamos:

    log(tan(x))sin(x)2sin(2x)+1sin(x)- \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)} - \frac{2}{\sin{\left(2 x \right)}} + \frac{1}{\sin{\left(x \right)}}


Respuesta:

log(tan(x))sin(x)2sin(2x)+1sin(x)- \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)} - \frac{2}{\sin{\left(2 x \right)}} + \frac{1}{\sin{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-200200
Primera derivada [src]
                        /       2   \                       
                        |1   tan (x)|                       
                      2*|- + -------|   /       2   \       
                        \2      2   /   \1 + tan (x)/*cos(x)
-log(tan(x))*sin(x) - --------------- + --------------------
                           tan(x)              tan(x)       
2(tan2(x)2+12)tan(x)+(tan2(x)+1)cos(x)tan(x)log(tan(x))sin(x)- \frac{2 \left(\frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)}{\tan{\left(x \right)}} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)}} - \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}
Segunda derivada [src]
                              2                                                              2                                
                 /       2   \                                                  /       2   \             /       2   \       
          2      \1 + tan (x)/                           /       2   \          \1 + tan (x)/ *cos(x)   2*\1 + tan (x)/*sin(x)
-2 - 2*tan (x) + -------------- - cos(x)*log(tan(x)) + 2*\1 + tan (x)/*cos(x) - --------------------- - ----------------------
                       2                                                                  2                     tan(x)        
                    tan (x)                                                            tan (x)                                
(tan2(x)+1)2cos(x)tan2(x)+(tan2(x)+1)2tan2(x)2(tan2(x)+1)sin(x)tan(x)+2(tan2(x)+1)cos(x)log(tan(x))cos(x)2tan2(x)2- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\tan^{2}{\left(x \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) \sin{\left(x \right)}}{\tan{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \log{\left(\tan{\left(x \right)} \right)} \cos{\left(x \right)} - 2 \tan^{2}{\left(x \right)} - 2
Tercera derivada [src]
                                                                                      3                  2                  2                                                  3                         2                                       
                                                                         /       2   \      /       2   \      /       2   \             /       2   \            /       2   \             /       2   \                                        
                       /       2   \            /       2   \          2*\1 + tan (x)/    4*\1 + tan (x)/    4*\1 + tan (x)/ *cos(x)   3*\1 + tan (x)/*cos(x)   2*\1 + tan (x)/ *cos(x)   3*\1 + tan (x)/ *sin(x)     /       2   \              
log(tan(x))*sin(x) - 6*\1 + tan (x)/*sin(x) - 4*\1 + tan (x)/*tan(x) - ---------------- + ---------------- - ----------------------- - ---------------------- + ----------------------- + ----------------------- + 4*\1 + tan (x)/*cos(x)*tan(x)
                                                                              3                tan(x)                 tan(x)                   tan(x)                      3                         2                                           
                                                                           tan (x)                                                                                      tan (x)                   tan (x)                                        
2(tan2(x)+1)3cos(x)tan3(x)2(tan2(x)+1)3tan3(x)+3(tan2(x)+1)2sin(x)tan2(x)4(tan2(x)+1)2cos(x)tan(x)+4(tan2(x)+1)2tan(x)6(tan2(x)+1)sin(x)+4(tan2(x)+1)cos(x)tan(x)3(tan2(x)+1)cos(x)tan(x)4(tan2(x)+1)tan(x)+log(tan(x))sin(x)\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \cos{\left(x \right)}}{\tan^{3}{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{3}{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\tan^{2}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\tan{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan{\left(x \right)}} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)}} - 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}
Gráfico
Derivada de y=cosx*ln(tgx)-ln(tgx/2)