Sr Examen

Derivada de ln(secx+tanx)

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

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Solución

Ha introducido [src]
log(sec(x) + tan(x))
log(tan(x)+sec(x))\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}
log(sec(x) + tan(x))
Solución detallada
  1. Sustituimos u=tan(x)+sec(x)u = \tan{\left(x \right)} + \sec{\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 ddx(tan(x)+sec(x))\frac{d}{d x} \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right):

    1. diferenciamos tan(x)+sec(x)\tan{\left(x \right)} + \sec{\left(x \right)} miembro por miembro:

      1. Reescribimos las funciones para diferenciar:

        sec(x)=1cos(x)\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}

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

      3. Según el principio, aplicamos: 1u\frac{1}{u} tenemos 1u2- \frac{1}{u^{2}}

      4. 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:

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

      5. Reescribimos las funciones para diferenciar:

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

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

    Como resultado de la secuencia de reglas:

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

  4. Simplificamos:

    1cos(x)\frac{1}{\cos{\left(x \right)}}


Respuesta:

1cos(x)\frac{1}{\cos{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-5050
Primera derivada [src]
       2                   
1 + tan (x) + sec(x)*tan(x)
---------------------------
      sec(x) + tan(x)      
tan2(x)+tan(x)sec(x)+1tan(x)+sec(x)\frac{\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1}{\tan{\left(x \right)} + \sec{\left(x \right)}}
Segunda derivada [src]
                                                                     2                         
                                        /       2                   \                          
   2             /       2   \          \1 + tan (x) + sec(x)*tan(x)/      /       2   \       
tan (x)*sec(x) + \1 + tan (x)/*sec(x) - ------------------------------ + 2*\1 + tan (x)/*tan(x)
                                               sec(x) + tan(x)                                 
-----------------------------------------------------------------------------------------------
                                        sec(x) + tan(x)                                        
2(tan2(x)+1)tan(x)+(tan2(x)+1)sec(x)+tan2(x)sec(x)(tan2(x)+tan(x)sec(x)+1)2tan(x)+sec(x)tan(x)+sec(x)\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)^{2}}{\tan{\left(x \right)} + \sec{\left(x \right)}}}{\tan{\left(x \right)} + \sec{\left(x \right)}}
Tercera derivada [src]
                                                                   3                                                                                                                                                             
               2                      /       2                   \                                /       2                   \ /   2             /       2   \            /       2   \       \                                
  /       2   \       3             2*\1 + tan (x) + sec(x)*tan(x)/         2    /       2   \   3*\1 + tan (x) + sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/     /       2   \              
2*\1 + tan (x)/  + tan (x)*sec(x) + -------------------------------- + 4*tan (x)*\1 + tan (x)/ - ------------------------------------------------------------------------------------------------ + 5*\1 + tan (x)/*sec(x)*tan(x)
                                                            2                                                                            sec(x) + tan(x)                                                                         
                                           (sec(x) + tan(x))                                                                                                                                                                     
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                         sec(x) + tan(x)                                                                                                         
2(tan2(x)+1)2+4(tan2(x)+1)tan2(x)+5(tan2(x)+1)tan(x)sec(x)+tan3(x)sec(x)3(2(tan2(x)+1)tan(x)+(tan2(x)+1)sec(x)+tan2(x)sec(x))(tan2(x)+tan(x)sec(x)+1)tan(x)+sec(x)+2(tan2(x)+tan(x)sec(x)+1)3(tan(x)+sec(x))2tan(x)+sec(x)\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + \tan^{3}{\left(x \right)} \sec{\left(x \right)} - \frac{3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)}{\tan{\left(x \right)} + \sec{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)^{3}}{\left(\tan{\left(x \right)} + \sec{\left(x \right)}\right)^{2}}}{\tan{\left(x \right)} + \sec{\left(x \right)}}
Gráfico
Derivada de ln(secx+tanx)