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Expresiones con funciones
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Derivada de la función/ tan(x)/ sin(cos(tan(x)^(3))^(2))

Derivada de sin(cos(tan(x)^(3))^(2))

Solución

Ha introducido [src]

   /   2/   3   \\
sin\cos \tan (x)//

$$\sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)}$$

sin(cos(tan(x)^3)^2)

Solución detallada

Sustituimos $u = \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)}$ .
La derivada del seno es igual al coseno:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)}$ :
1. Sustituimos $u = \cos{\left(\tan^{3}{\left(x \right)} \right)}$ .
2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(\tan^{3}{\left(x \right)} \right)}$ :
  1. Sustituimos $u = \tan^{3}{\left(x \right)}$ .
  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 $\frac{d}{d x} \tan^{3}{\left(x \right)}$ :
    1. Sustituimos $u = \tan{\left(x \right)}$ .
    2. Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
      1. Reescribimos las funciones para diferenciar:
        
        $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      2. Se aplica la regla de la derivada parcial:
        
        $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
        
        $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
        
        Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
        
        La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
        
        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 la secuencia de reglas:
      
      $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    Como resultado de la secuencia de reglas:
    
    $- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Como resultado de la secuencia de reglas:
  
  $- \frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Como resultado de la secuencia de reglas:

$- \frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Simplificamos:

$- \frac{6 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

Respuesta:

$- \frac{6 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

      2    /         2   \    /   2/   3   \\    /   3   \    /   3   \
-2*tan (x)*\3 + 3*tan (x)/*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/

$$- 2 \left(3 \tan^{2}{\left(x \right)} + 3\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}$$

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                 /               2                                                               2                                                                                                                                                                               2                                                                                                                       2                                                                         2                                                                                                                                                                                                                           2                                                                          2                                                                        2                                                      \
   /       2   \ |  /       2   \     /   2/   3   \\    /   3   \    /   3   \     /       2   \     2/   3   \    3       /   2/   3   \\        2/   3   \    5    /       2   \    /   2/   3   \\        4       /   2/   3   \\    /   3   \    /   3   \     /       2   \     2/   3   \    3       /   2/   3   \\        2/   3   \    5    /       2   \    /   2/   3   \\      /       2   \     3/   3   \    6       /   2/   3   \\    /   3   \      /       2   \     2/   3   \    2/   3   \    3       /   2/   3   \\         2/   3   \    2/   3   \    5    /       2   \    /   2/   3   \\        2    /       2   \    /   2/   3   \\    /   3   \    /   3   \      /       2   \     3/   3   \    3/   3   \    6       /   2/   3   \\      /       2   \     6       /   2/   3   \\    /   3   \    /   3   \      /       2   \     3/   3   \    6       /   3   \    /   2/   3   \\|
12*\1 + tan (x)/*\- \1 + tan (x)/ *cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ - 9*\1 + tan (x)/ *cos \tan (x)/*tan (x)*cos\cos \tan (x)// - 9*cos \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 2*tan (x)*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 9*\1 + tan (x)/ *sin \tan (x)/*tan (x)*cos\cos \tan (x)// + 9*sin \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 27*\1 + tan (x)/ *cos \tan (x)/*tan (x)*sin\cos \tan (x)//*sin\tan (x)/ - 18*\1 + tan (x)/ *cos \tan (x)/*sin \tan (x)/*tan (x)*sin\cos \tan (x)// - 18*cos \tan (x)/*sin \tan (x)/*tan (x)*\1 + tan (x)/*sin\cos \tan (x)// - 7*tan (x)*\1 + tan (x)/*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 18*\1 + tan (x)/ *cos \tan (x)/*sin \tan (x)/*tan (x)*cos\cos \tan (x)// + 18*\1 + tan (x)/ *tan (x)*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 27*\1 + tan (x)/ *sin \tan (x)/*tan (x)*cos\tan (x)/*sin\cos \tan (x)///

$$12 \left(\tan^{2}{\left(x \right)} + 1\right) \left(27 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} - 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 27 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} + 9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \tan^{3}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 18 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} + 9 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \tan^{5}{\left(x \right)} - 7 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} - 2 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{4}{\left(x \right)}\right)$$

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Gráfico

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Expresiones idénticas

Expresiones con funciones

Derivada de sin(cos(tan(x)^(3))^(2))

Gráfico:

Definida a trozos:

Solución