Derivada de y=-(cosx)ln(secx+tanx)

Solución

Ha introducido [src]

-cos(x)*log(sec(x) + tan(x))

$$\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \left(- \cos{\left(x \right)}\right)$$

(-cos(x))*log(sec(x) + tan(x))

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = - \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
  1. La derivada del coseno es igual a menos el seno:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Entonces, como resultado: $\sin{\left(x \right)}$
$g{\left(x \right)} = \log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = \tan{\left(x \right)} + \sec{\left(x \right)}$ .
2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right)$ :
  1. diferenciamos $\tan{\left(x \right)} + \sec{\left(x \right)}$ miembro por miembro:
    1. Reescribimos las funciones para diferenciar:
      
      $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
    2. Sustituimos $u = \cos{\left(x \right)}$ .
    3. Según el principio, aplicamos: $\frac{1}{u}$ tenemos $- \frac{1}{u^{2}}$
    4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(x \right)}$ :
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Como resultado de la secuencia de reglas:
      
      $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    5. Reescribimos las funciones para diferenciar:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    6. Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    Como resultado de: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Como resultado de la secuencia de reglas:
  
  $\frac{\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}}{\tan{\left(x \right)} + \sec{\left(x \right)}}$
Como resultado de: $- \frac{\left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + \sec{\left(x \right)}} + \log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \sin{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                              /       2                   \       
                              \1 + tan (x) + sec(x)*tan(x)/*cos(x)
log(sec(x) + tan(x))*sin(x) - ------------------------------------
                                        sec(x) + tan(x)

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

Simplificar

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)|*cos(x)     /       2                   \       
                              \                                               sec(x) + tan(x)                                 /          2*\1 + tan (x) + sec(x)*tan(x)/*sin(x)
cos(x)*log(sec(x) + tan(x)) - -------------------------------------------------------------------------------------------------------- + --------------------------------------
                                                                          sec(x) + tan(x)                                                           sec(x) + tan(x)

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

Simplificar

Tercera derivada [src]

                               /                                                                   3                                                                                                                                                             \                                                                                                                                                             
                               |               2                      /       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                   \                          |       
                               |2*\1 + tan (x)/  + tan (x)*sec(x) + -------------------------------- + 4*tan (x)*\1 + tan (x)/ - ------------------------------------------------------------------------------------------------ + 5*\1 + tan (x)/*sec(x)*tan(x)|*cos(x)                                              |   2             /       2   \          \1 + tan (x) + sec(x)*tan(x)/      /       2   \       |       
                               |                                                            2                                                                            sec(x) + tan(x)                                                                         |            /       2                   \          3*|tan (x)*sec(x) + \1 + tan (x)/*sec(x) - ------------------------------ + 2*\1 + tan (x)/*tan(x)|*sin(x)
                               \                                           (sec(x) + tan(x))                                                                                                                                                                     /          3*\1 + tan (x) + sec(x)*tan(x)/*cos(x)     \                                               sec(x) + tan(x)                                 /       
-log(sec(x) + tan(x))*sin(x) - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------- + ----------------------------------------------------------------------------------------------------------
                                                                                                                                            sec(x) + tan(x)                                                                                                                            sec(x) + tan(x)                                                            sec(x) + tan(x)

$$- \log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \sin{\left(x \right)} + \frac{3 \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + \sec{\left(x \right)}} + \frac{3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)^{2}}{\tan{\left(x \right)} + \sec{\left(x \right)}}\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + \sec{\left(x \right)}} - \frac{\left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + \tan^{3}{\left(x \right)} \sec{\left(x \right)} - \frac{3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)}{\tan{\left(x \right)} + \sec{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)^{3}}{\left(\tan{\left(x \right)} + \sec{\left(x \right)}\right)^{2}}\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + \sec{\left(x \right)}}$$

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Gráfico

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Derivada de y=-(cosx)ln(secx+tanx)

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