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Derivada de:
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Expresiones idénticas
tan^- uno (secx+tanx)
tangente de en el grado menos 1(secx más tangente de x)
tangente de en el grado menos uno (secx más tangente de x)
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tan-1secx+tanx
tan^-1secx+tanx
Expresiones semejantes
tan^-1(secx-tanx)
tan^+1(secx+tanx)
Expresiones con funciones
Tangente tan
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tan(x)^3
secx
secx/tanx

Derivada de la función/ secx+tanx/ tan^-1(secx+tanx)

Derivada de tan^-1(secx+tanx)

Solución

Ha introducido [src]

         1          
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tan(sec(x) + tan(x))

$$\frac{1}{\tan{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}}$$

1/tan(sec(x) + tan(x))

Solución detallada

Sustituimos $u = \tan{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$ .
Según el principio, aplicamos: $\frac{1}{u}$ tenemos $- \frac{1}{u^{2}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$ :
1. Reescribimos las funciones para diferenciar:
  
  $\tan{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} = \frac{\sin{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}}{\cos{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}}$
2. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$ y $g{\left(x \right)} = \cos{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. Sustituimos $u = \tan{\left(x \right)} + \sec{\left(x \right)}$ .
  2. La derivada del seno es igual al coseno:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  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 $\frac{d}{d x} \cos{\left(x \right)}$ :
        
        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. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      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:
    
    $\left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = \tan{\left(x \right)} + \sec{\left(x \right)}$ .
  2. La derivada del coseno es igual a menos el seno:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  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. La derivada de la secante es igual a la secante por tangente:
        
        $\frac{d}{d x} \sec{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$
      2. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      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:
    
    $- \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \sin{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{\left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \sin^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} + \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}}{\cos^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}}$
Como resultado de la secuencia de reglas:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                            /                                                    2                                                                                                           2                              3                                                                                                                                      3                                                                                                                                                           \
                            |                                 3     /       2   \       3                  2    /       2   \     /       2   \                   /       2                 \  /       2                   \      /       2                   \ /   2             /       2   \            /       2   \       \      /       2                   \  /       2                 \     /       2                 \ /       2                   \ /   2             /       2   \            /       2   \       \|
/       2                 \ |    /       2                   \    2*\1 + tan (x)/  + tan (x)*sec(x) + 4*tan (x)*\1 + tan (x)/ + 5*\1 + tan (x)/*sec(x)*tan(x)   6*\1 + tan (sec(x) + tan(x))/ *\1 + tan (x) + sec(x)*tan(x)/    6*\1 + tan (x) + sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/   10*\1 + tan (x) + sec(x)*tan(x)/ *\1 + tan (sec(x) + tan(x))/   6*\1 + tan (sec(x) + tan(x))/*\1 + tan (x) + sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/|
\1 + tan (sec(x) + tan(x))/*|- 4*\1 + tan (x) + sec(x)*tan(x)/  - ------------------------------------------------------------------------------------------- - ------------------------------------------------------------- - ------------------------------------------------------------------------------------------------ + ------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------|
                            |                                                                           2                                                                              4                                                                              tan(sec(x) + tan(x))                                                                2                                                                                              3                                                                     |
                            \                                                                        tan (sec(x) + tan(x))                                                          tan (sec(x) + tan(x))                                                                                                                                              tan (sec(x) + tan(x))                                                                          tan (sec(x) + tan(x))                                                    /

$$\left(\tan^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} + 1\right) \left(- \frac{6 \left(\tan^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} + 1\right)^{2} \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}} + \frac{6 \left(\tan^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} + 1\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)}{\tan^{3}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}} + \frac{10 \left(\tan^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)^{3}}{\tan^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}} - \frac{6 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)}{\tan{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}} - 4 \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)^{3} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + \tan^{3}{\left(x \right)} \sec{\left(x \right)}}{\tan^{2}{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}}\right)$$

Simplificar

Gráfico

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Expresiones con funciones

Derivada de tan^-1(secx+tanx)

Gráfico:

Definida a trozos:

Solución