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Derivada de:
Derivada de (3x^4-x)^7
Derivada de (3x+6)^2
Derivada de (1+cos(x))/(1-cos(x))
Derivada de (2*x-9)/(x-5)
Expresiones idénticas
y=sin(tgx^ tres)^ tres
y es igual a seno de (tgx al cubo ) al cubo
y es igual a seno de (tgx en el grado tres) en el grado tres
y=sin(tgx3)3
y=sintgx33
y=sin(tgx³)³
y=sin(tgx en el grado 3) en el grado 3
y=sintgx^3^3

Derivada de la función/ y=sin/ y=sin(tgx^3)^3

Derivada de y=sin(tgx^3)^3

Solución

Ha introducido [src]

   3/   3   \
sin \tan (x)/

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

sin(tan(x)^3)^3

Solución detallada

Sustituimos $u = \sin{\left(\tan^{3}{\left(x \right)} \right)}$ .
Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(\tan^{3}{\left(x \right)} \right)}$ :
1. Sustituimos $u = \tan^{3}{\left(x \right)}$ .
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 $\frac{d}{d x} \tan^{3}{\left(x \right)}$ :
  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 $\frac{d}{d x} f{\left(x \right)}$ :
      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 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:
  
  $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Como resultado de la secuencia de reglas:

$\frac{9 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Simplificamos:

$\frac{9 \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

Respuesta:

$\frac{9 \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

  /       2   \ /       2/   3   \    3    /       2   \        2       /   3   \    /   3   \     /       2   \    /   3   \    /   3   \        2/   3   \    3    /       2   \\    /   3   \       
9*\1 + tan (x)/*\- 3*sin \tan (x)/*tan (x)*\1 + tan (x)/ + 2*tan (x)*cos\tan (x)/*sin\tan (x)/ + 2*\1 + tan (x)/*cos\tan (x)/*sin\tan (x)/ + 6*cos \tan (x)/*tan (x)*\1 + tan (x)//*sin\tan (x)/*tan(x)

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

Simplificar

Tercera derivada [src]

                /                  2                                                                                 2                                                                                     2                                         2                                                                                                            2                                                                                         \
  /       2   \ |     /       2   \     3/   3   \    3            3/   3   \    5    /       2   \     /       2   \     2/   3   \    /   3   \        2/   3   \    4       /   3   \      /       2   \     3/   3   \    6         /       2   \     2/   3   \    6       /   3   \         2/   3   \    2    /       2   \    /   3   \      /       2   \     2/   3   \    3       /   3   \         2/   3   \    5    /       2   \    /   3   \|
9*\1 + tan (x)/*\- 18*\1 + tan (x)/ *sin \tan (x)/*tan (x) - 18*sin \tan (x)/*tan (x)*\1 + tan (x)/ + 2*\1 + tan (x)/ *sin \tan (x)/*cos\tan (x)/ + 4*sin \tan (x)/*tan (x)*cos\tan (x)/ + 18*\1 + tan (x)/ *cos \tan (x)/*tan (x) - 63*\1 + tan (x)/ *sin \tan (x)/*tan (x)*cos\tan (x)/ + 14*sin \tan (x)/*tan (x)*\1 + tan (x)/*cos\tan (x)/ + 36*\1 + tan (x)/ *cos \tan (x)/*tan (x)*sin\tan (x)/ + 36*cos \tan (x)/*tan (x)*\1 + tan (x)/*sin\tan (x)//

$$9 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 63 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} + 36 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} - 18 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} + 14 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)} + 36 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} + 4 \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{4}{\left(x \right)}\right)$$

Simplificar

Gráfico

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Expresiones idénticas

Derivada de y=sin(tgx^3)^3

Gráfico:

Definida a trozos:

Solución