Derivada de (secx+tanx)(secx-tanx)

Solución

Ha introducido [src]

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

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

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

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \tan{\left(x \right)} + \sec{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = - \tan{\left(x \right)} + \sec{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
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. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
    1. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
    Entonces, como resultado: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  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: $\left(- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) + \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \left(- \tan{\left(x \right)} + \sec{\left(x \right)}\right)$
Simplificamos:

$0$

Respuesta:

$0$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

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Gráfico

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Derivada de (secx+tanx)(secx-tanx)

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