Derivada de y=tg3tgx+3tgtgx

Solución

Ha introducido [src]

tan(3*tan(x)) + 3*tan(tan(x))

$$3 \tan{\left(\tan{\left(x \right)} \right)} + \tan{\left(3 \tan{\left(x \right)} \right)}$$

tan(3*tan(x)) + 3*tan(tan(x))

Solución detallada

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

$\frac{3 \left(\frac{1}{\cos^{2}{\left(3 \tan{\left(x \right)} \right)}} + \frac{1}{\cos^{2}{\left(\tan{\left(x \right)} \right)}}\right)}{\cos^{2}{\left(x \right)}}$

Respuesta:

$\frac{3 \left(\frac{1}{\cos^{2}{\left(3 \tan{\left(x \right)} \right)}} + \frac{1}{\cos^{2}{\left(\tan{\left(x \right)} \right)}}\right)}{\cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

/       2          \ /         2   \     /       2   \ /       2        \
\1 + tan (3*tan(x))/*\3 + 3*tan (x)/ + 3*\1 + tan (x)/*\1 + tan (tan(x))/

$$3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) + \left(3 \tan^{2}{\left(x \right)} + 3\right) \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right)$$

Simplificar

Segunda derivada [src]

  /       2   \ //       2          \          /       2        \          /       2   \ /       2        \                 /       2   \ /       2          \              \
6*\1 + tan (x)/*\\1 + tan (3*tan(x))/*tan(x) + \1 + tan (tan(x))/*tan(x) + \1 + tan (x)/*\1 + tan (tan(x))/*tan(tan(x)) + 3*\1 + tan (x)/*\1 + tan (3*tan(x))/*tan(3*tan(x))/

$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) \tan{\left(\tan{\left(x \right)} \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right) \tan{\left(3 \tan{\left(x \right)} \right)} + \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right) \tan{\left(x \right)}\right)$$

Simplificar

Tercera derivada [src]

                /             2                   2                                                                                                                                                          2                     2                  2                                                   2                                                                                                                                                         \
  /       2   \ |/       2   \  /       2        \    /       2   \ /       2          \   /       2   \ /       2        \        2    /       2          \        2    /       2        \     /       2   \  /       2          \      /       2   \     2         /       2        \      /       2   \     2           /       2          \     /       2   \ /       2        \                         /       2   \ /       2          \                     |
6*\1 + tan (x)/*\\1 + tan (x)/ *\1 + tan (tan(x))/  + \1 + tan (x)/*\1 + tan (3*tan(x))/ + \1 + tan (x)/*\1 + tan (tan(x))/ + 2*tan (x)*\1 + tan (3*tan(x))/ + 2*tan (x)*\1 + tan (tan(x))/ + 9*\1 + tan (x)/ *\1 + tan (3*tan(x))/  + 2*\1 + tan (x)/ *tan (tan(x))*\1 + tan (tan(x))/ + 18*\1 + tan (x)/ *tan (3*tan(x))*\1 + tan (3*tan(x))/ + 6*\1 + tan (x)/*\1 + tan (tan(x))/*tan(x)*tan(tan(x)) + 18*\1 + tan (x)/*\1 + tan (3*tan(x))/*tan(x)*tan(3*tan(x))/

$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{2}{\left(\tan{\left(x \right)} \right)} + 9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right) \tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) \tan{\left(x \right)} \tan{\left(\tan{\left(x \right)} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) + 18 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right) \tan{\left(x \right)} \tan{\left(3 \tan{\left(x \right)} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right) + 2 \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \left(\tan^{2}{\left(3 \tan{\left(x \right)} \right)} + 1\right) \tan^{2}{\left(x \right)}\right)$$

Simplificar

Gráfico

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Derivada de y=tg3tgx+3tgtgx

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