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Derivada de:
Derivada de x^-(4/5)
Derivada de x^-8
Derivada de x^2/lnx
Derivada de √x+2
Expresiones idénticas
y=x^3ln^ dos (sin^2x-tg^2x)
y es igual a x al cubo ln al cuadrado ( seno de al cuadrado x menos tg al cuadrado x)
y es igual a x al cubo ln en el grado dos ( seno de al cuadrado x menos tg al cuadrado x)
y=x3ln2(sin2x-tg2x)
y=x3ln2sin2x-tg2x
y=x³ln²(sin²x-tg²x)
y=x en el grado 3ln en el grado 2(sin en el grado 2x-tg en el grado 2x)
y=x^3ln^2sin^2x-tg^2x
Expresiones semejantes
y=x^3ln^2(sin^2x+tg^2x)
Expresiones con funciones
Seno sin
sin*x^2
sin(x)^(6)
sin(x)^(5*x)
sin(x+2)
sin(x^3-x)/(x^3-x)+sin(x^3-x)

Derivada de la función/ y=x^3/ tg^2x/ sin^2/ y=x^3ln^2(sin^2x-tg^2x)

Derivada de y=x^3ln^2(sin^2x-tg^2x)

Solución

Ha introducido [src]

 3    2/   2         2   \
x *log \sin (x) - tan (x)/

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

x^3*log(sin(x)^2 - tan(x)^2)^2

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x^{3}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x^{3}$ tenemos $3 x^{2}$
$g{\left(x \right)} = \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}^{2}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}$ .
2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}$ :
  1. Sustituimos $u = \sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}$ .
  2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}\right)$ :
    1. diferenciamos $\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}$ miembro por miembro:
      1. Sustituimos $u = \sin{\left(x \right)}$ .
      2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
      3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
        
        La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        Como resultado de la secuencia de reglas:
        
        $2 \sin{\left(x \right)} \cos{\left(x \right)}$
      4. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
        
        Sustituimos $u = \tan{\left(x \right)}$ .
        
        Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
        
        Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
        
        Reescribimos las funciones para diferenciar:
        
        $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
        
        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:
        
        $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
        
        Entonces, como resultado: $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      Como resultado de: $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    Como resultado de la secuencia de reglas:
    
    $\frac{- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}}$
  Como resultado de la secuencia de reglas:
  
  $\frac{2 \left(- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}}$
Como resultado de: $\frac{2 x^{3} \left(- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} + 3 x^{2} \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}^{2}$
Simplificamos:

$\frac{x^{2} \left(\frac{4 x \cos^{3}{\left(x \right)}}{\sin{\left(x \right)}} - \frac{4 x \tan{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 3 \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)} \sin^{2}{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} - 1}$

Respuesta:

$\frac{x^{2} \left(\frac{4 x \cos^{3}{\left(x \right)}}{\sin{\left(x \right)}} - \frac{4 x \tan{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 3 \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)} \sin^{2}{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} - 1}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

    /                                 /                                                                                                                                2                                           2                       \                                                                     \
    |                                 |/             2                                              \                            //       2   \                       \      //       2   \                       \     /   2         2   \|                                                                     |
    |                               2 ||/       2   \       2         2           2    /       2   \|    /   2         2   \   2*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/    2*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/ *log\sin (x) - tan (x)/|                                                                     |
    |                            2*x *|\\1 + tan (x)/  + sin (x) - cos (x) + 2*tan (x)*\1 + tan (x)//*log\sin (x) - tan (x)/ - ----------------------------------------- + ----------------------------------------------------------------|                                                                     |
    |                                 |                                                                                                       2         2                                            2         2                           |        //       2   \                       \    /   2         2   \|
    |     2/   2         2   \        \                                                                                                    sin (x) - tan (x)                                      sin (x) - tan (x)                        /   12*x*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/*log\sin (x) - tan (x)/|
2*x*|3*log \sin (x) - tan (x)/ - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------|
    |                                                                                                                            2         2                                                                                                                              2         2                            |
    \                                                                                                                         sin (x) - tan (x)                                                                                                                        sin (x) - tan (x)                         /

$$2 x \left(- \frac{2 x^{2} \left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right)^{2} \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} - \frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right)^{2}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} + \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}\right)}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} - \frac{12 x \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} + 3 \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}^{2}\right)$$

Simplificar

Tercera derivada [src]

  /                                                                                                                                                                                                                                                /                                          3                                                                                                                                         /             2                                              \                                           3                                                                   /             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   \|      3 |  6*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/      |   3    /       2   \                     /       2   \        |    /   2         2   \   3*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/*\\1 + tan (x)/  + sin (x) - cos (x) + 2*tan (x)*\1 + tan (x)//   4*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/ *log\sin (x) - tan (x)/   3*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/*\\1 + tan (x)/  + sin (x) - cos (x) + 2*tan (x)*\1 + tan (x)//*log\sin (x) - tan (x)/|                                                                     |
  |                                2 ||/       2   \       2         2           2    /       2   \|    /   2         2   \   2*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/    2*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/ *log\sin (x) - tan (x)/|   4*x *|- ----------------------------------------- + 2*\tan (x)*\1 + tan (x)/ + cos(x)*sin(x) + 2*\1 + tan (x)/ *tan(x)/*log\sin (x) - tan (x)/ - ------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------|                                                                     |
  |                            18*x *|\\1 + tan (x)/  + sin (x) - cos (x) + 2*tan (x)*\1 + tan (x)//*log\sin (x) - tan (x)/ - ----------------------------------------- + ----------------------------------------------------------------|        |                                2                                                                                                                                                        2         2                                                                                          2                                                                                  2         2                                                          |                                                                     |
  |                                  |                                                                                                       2         2                                            2         2                           |        |             /   2         2   \                                                                                                                                                      sin (x) - tan (x)                                                                    /   2         2   \                                                                                sin (x) - tan (x)                                                       |        //       2   \                       \    /   2         2   \|
  |     2/   2         2   \         \                                                                                                    sin (x) - tan (x)                                      sin (x) - tan (x)                        /        \             \sin (x) - tan (x)/                                                                                                                                                                                                                                           \sin (x) - tan (x)/                                                                                                                                                        /   36*x*\\1 + tan (x)/*tan(x) - cos(x)*sin(x)/*log\sin (x) - tan (x)/|
2*|3*log \sin (x) - tan (x)/ - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------|
  |                                                                                                                            2         2                                                                                                                                                                                                                                                                                                                             2         2                                                                                                                                                                                                                                                       2         2                            |
  \                                                                                                                         sin (x) - tan (x)                                                                                                                                                                                                                                                                                                                       sin (x) - tan (x)                                                                                                                                                                                                                                                 sin (x) - tan (x)                         /

$$2 \left(- \frac{4 x^{3} \left(\frac{4 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right)^{3} \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}\right)^{2}} - \frac{6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right)^{3}}{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}\right)^{2}} + \frac{3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} - \frac{3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} + 2 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}\right)}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} - \frac{18 x^{2} \left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right)^{2} \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} - \frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right)^{2}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} + \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}\right)}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} - \frac{36 x \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}} + 3 \log{\left(\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \right)}^{2}\right)$$

Simplificar

Gráfico

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de y=x^3ln^2(sin^2x-tg^2x)

Gráfico:

Definida a trozos:

Solución