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Derivada de:
Derivada de x*e
Derivada de -2x
Derivada de e^-1
Derivada de 8/x
Expresiones idénticas
((x+ uno)*x/sin(x)^ dos)^(uno / tres)
((x más 1) multiplicar por x dividir por seno de (x) al cuadrado ) en el grado (1 dividir por 3)
((x más uno) multiplicar por x dividir por seno de (x) en el grado dos) en el grado (uno dividir por tres)
((x+1)*x/sin(x)2)(1/3)
x+1*x/sinx21/3
((x+1)*x/sin(x)²)^(1/3)
((x+1)*x/sin(x) en el grado 2) en el grado (1/3)
((x+1)x/sin(x)^2)^(1/3)
((x+1)x/sin(x)2)(1/3)
x+1x/sinx21/3
x+1x/sinx^2^1/3
((x+1)*x dividir por sin(x)^2)^(1 dividir por 3)
Expresiones semejantes
((x-1)*x/sin(x)^2)^(1/3)
((x+1)*x/sinx^2)^(1/3)
Expresiones con funciones
Seno sin
sin(x)/x^2
sin(3*x+2)
sin(√x)
sin^2*x
sin(x)+3

Derivada de la función/ (x)^2/ (x+1)/ sin(x)/ ((x+1)*x/sin(x)^2)^(1/3)

Derivada de ((x+1)*x/sin(x)^2)^(1/3)

Solución

Ha introducido [src]

     ___________
    / (x + 1)*x 
   /  --------- 
3 /       2     
\/     sin (x)

$$\sqrt[3]{\frac{x \left(x + 1\right)}{\sin^{2}{\left(x \right)}}}$$

(((x + 1)*x)/sin(x)^2)^(1/3)

Solución detallada

Sustituimos $u = \frac{x \left(x + 1\right)}{\sin^{2}{\left(x \right)}}$ .
Según el principio, aplicamos: $\sqrt[3]{u}$ tenemos $\frac{1}{3 u^{\frac{2}{3}}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x \left(x + 1\right)}{\sin^{2}{\left(x \right)}}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x \left(x + 1\right)$ y $g{\left(x \right)} = \sin^{2}{\left(x \right)}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. Según el principio, aplicamos: $x$ tenemos $1$
    $g{\left(x \right)} = x + 1$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. diferenciamos $x + 1$ miembro por miembro:
      1. La derivada de una constante $1$ es igual a cero.
      2. Según el principio, aplicamos: $x$ tenemos $1$
      Como resultado de: $1$
    Como resultado de: $2 x + 1$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  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 $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. 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:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- 2 x \left(x + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(2 x + 1\right) \sin^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$
Como resultado de la secuencia de reglas:

$\frac{- 2 x \left(x + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(2 x + 1\right) \sin^{2}{\left(x \right)}}{3 \left(\frac{x \left(x + 1\right)}{\sin^{2}{\left(x \right)}}\right)^{\frac{2}{3}} \sin^{4}{\left(x \right)}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

     ___________                                         
    / x*(x + 1)     2    / 1 + 2*x    2*x*(x + 1)*cos(x)\
   /  --------- *sin (x)*|--------- - ------------------|
3 /       2              |     2               3        |
\/     sin (x)           \3*sin (x)       3*sin (x)     /
---------------------------------------------------------
                        x*(x + 1)

$$\frac{\sqrt[3]{\frac{x \left(x + 1\right)}{\sin^{2}{\left(x \right)}}} \left(- \frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}} + \frac{2 x + 1}{3 \sin^{2}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{x \left(x + 1\right)}$$

Simplificar

Segunda derivada [src]

                 /               2*x*(1 + x)*cos(x)              2*x*(1 + x)*cos(x)                                                                        /           2*x*(1 + x)*cos(x)\                                /          2*x*(1 + x)*cos(x)\ /           2*x*(1 + x)*cos(x)\\
     ___________ |    -1 - 2*x + ------------------   -1 - 2*x + ------------------                                                                      2*|-1 - 2*x + ------------------|*cos(x)          2              |1 + 2*x - ------------------|*|-1 - 2*x + ------------------||
    / x*(1 + x)  |2                    sin(x)                          sin(x)         2*x*(1 + x)   2*x*cos(x)   2*(1 + x)*cos(x)   2*(1 + 2*x)*cos(x)     \                 sin(x)      /          2*x*cos (x)*(1 + x)   \                sin(x)      / \                 sin(x)      /|
   /  --------- *|- + ----------------------------- + ----------------------------- + ----------- - ---------- - ---------------- - ------------------ - ---------------------------------------- + ------------------- - --------------------------------------------------------------|
3 /       2      |3                3*x                          3*(1 + x)                  3         3*sin(x)        3*sin(x)            3*sin(x)                        3*sin(x)                            2                                     9*x*(1 + x)                          |
\/     sin (x)   \                                                                                                                                                                                        sin (x)                                                                       /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                        x*(1 + x)

$$\frac{\sqrt[3]{\frac{x \left(x + 1\right)}{\sin^{2}{\left(x \right)}}} \left(\frac{2 x \left(x + 1\right)}{3} + \frac{2 x \left(x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \cos{\left(x \right)}}{3 \sin{\left(x \right)}} - \frac{2 \left(x + 1\right) \cos{\left(x \right)}}{3 \sin{\left(x \right)}} - \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{3 \sin{\left(x \right)}} - \frac{2 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right) \cos{\left(x \right)}}{3 \sin{\left(x \right)}} + \frac{2}{3} + \frac{\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1}{3 \left(x + 1\right)} + \frac{\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1}{3 x} - \frac{\left(- \frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x + 1\right) \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right)}{9 x \left(x + 1\right)}\right)}{x \left(x + 1\right)}$$

Simplificar

Tercera derivada [src]

                 /                       /                                                                      2           \     /                                                                      2           \                                                                                                                                                                                                                     /                                                                      2           \                                                                                       /                                                       2           \                                                                                                                                                                                                                                                                                                                              /                                                                      2           \                                                                          \
                 |                       |                x*cos(x)   (1 + x)*cos(x)   (1 + 2*x)*cos(x)   3*x*cos (x)*(1 + x)|     |                x*cos(x)   (1 + x)*cos(x)   (1 + 2*x)*cos(x)   3*x*cos (x)*(1 + x)|                                                                                                                                                                                                                     |                x*cos(x)   (1 + x)*cos(x)   (1 + 2*x)*cos(x)   3*x*cos (x)*(1 + x)|                                                       /           2*x*(1 + x)*cos(x)\ |                2*x*cos(x)   2*(1 + x)*cos(x)   3*x*cos (x)*(1 + x)|                                 2                                                                                                                                                                                                                                                             /          2*x*(1 + x)*cos(x)\ |                x*cos(x)   (1 + x)*cos(x)   (1 + 2*x)*cos(x)   3*x*cos (x)*(1 + x)|                                                                          |
                 |                     4*|1 + x*(1 + x) - -------- - -------------- - ---------------- + -------------------|   4*|1 + x*(1 + x) - -------- - -------------- - ---------------- + -------------------|     /           2*x*(1 + x)*cos(x)\     /           2*x*(1 + x)*cos(x)\                                                             /           2*x*(1 + x)*cos(x)\        2    /           2*x*(1 + x)*cos(x)\   8*|1 + x*(1 + x) - -------- - -------------- - ---------------- + -------------------|*cos(x)                                              2*|-1 - 2*x + ------------------|*|1 + x*(1 + x) - ---------- - ---------------- + -------------------|   /          2*x*(1 + x)*cos(x)\  /           2*x*(1 + x)*cos(x)\   /          2*x*(1 + x)*cos(x)\ /           2*x*(1 + x)*cos(x)\   /          2*x*(1 + x)*cos(x)\ /           2*x*(1 + x)*cos(x)\     /           2*x*(1 + x)*cos(x)\            /           2*x*(1 + x)*cos(x)\          4*|1 + 2*x - ------------------|*|1 + x*(1 + x) - -------- - -------------- - ---------------- + -------------------|     /          2*x*(1 + x)*cos(x)\ /           2*x*(1 + x)*cos(x)\       |
     ___________ |                       |                 sin(x)        sin(x)            sin(x)                 2         |     |                 sin(x)        sin(x)            sin(x)                 2         |   2*|-1 - 2*x + ------------------|   2*|-1 - 2*x + ------------------|        2                       2           2              2*|-1 - 2*x + ------------------|   2*cos (x)*|-1 - 2*x + ------------------|     |                 sin(x)        sin(x)            sin(x)                 2         |                 3                                     \                 sin(x)      / |                  sin(x)          sin(x)                 2         |   |1 + 2*x - ------------------| *|-1 - 2*x + ------------------|   |1 + 2*x - ------------------|*|-1 - 2*x + ------------------|   |1 + 2*x - ------------------|*|-1 - 2*x + ------------------|   4*|-1 - 2*x + ------------------|*cos(x)   4*|-1 - 2*x + ------------------|*cos(x)     \                sin(x)      / |                 sin(x)        sin(x)            sin(x)                 2         |   2*|1 + 2*x - ------------------|*|-1 - 2*x + ------------------|*cos(x)|
    / x*(1 + x)  |4   8*x   4*cos(x)     \                                                                     sin (x)      /     \                                                                     sin (x)      /     \                 sin(x)      /     \                 sin(x)      /   2*cos (x)*(1 + 2*x)   4*x*cos (x)   4*cos (x)*(1 + x)     \                 sin(x)      /             \                 sin(x)      /     \                                                                     sin (x)      /          8*x*cos (x)*(1 + x)   4*x*(1 + x)*cos(x)                                     \                                                      sin (x)      /   \                sin(x)      /  \                 sin(x)      /   \                sin(x)      / \                 sin(x)      /   \                sin(x)      / \                 sin(x)      /     \                 sin(x)      /            \                 sin(x)      /                                           \                                                                     sin (x)      /     \                sin(x)      / \                 sin(x)      /       |
   /  --------- *|- + --- - -------- - -------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------- - --------------------------------- - --------------------------------- + ------------------- + ----------- + ----------------- - --------------------------------- - ----------------------------------------- + --------------------------------------------------------------------------------------------- - ------------------- - ------------------ - ------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------- + -------------------------------------------------------------- + -------------------------------------------------------------- + ---------------------------------------- + ---------------------------------------- + --------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------|
3 /       2      |3    3     sin(x)                                             3*x                                                                                   3*(1 + x)                                                            2                                      2                       2                 2                2                      3*x*(1 + x)                                   2                                                                 3*sin(x)                                                      3                  sin(x)                                                       9*x*(1 + x)                                                                              2        2                                                               2                                                        2                                                   3*x*sin(x)                              3*(1 + x)*sin(x)                                                                    9*x*(1 + x)                                                                                   3*x*(1 + x)*sin(x)                          |
\/     sin (x)   \                                                                                                                                                                                                                      3*x                              3*(1 + x)                     sin (x)           sin (x)          sin (x)                                                            3*sin (x)                                                                                                                         sin (x)                                                                                                                                                                 27*x *(1 + x)                                                     3*x*(1 + x)                                                      3*x *(1 + x)                                                                                                                                                                                                                                                                                                                 /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       x*(1 + x)

$$\frac{\sqrt[3]{\frac{x \left(x + 1\right)}{\sin^{2}{\left(x \right)}}} \left(- \frac{4 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{8 x \left(x + 1\right) \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{8 x}{3} + \frac{4 x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \left(x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(2 x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right) \cos^{2}{\left(x \right)}}{3 \sin^{2}{\left(x \right)}} + \frac{8 \left(x \left(x + 1\right) + \frac{3 x \left(x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 1\right) \cos{\left(x \right)}}{3 \sin{\left(x \right)}} + \frac{4}{3} - \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{4 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right) \cos{\left(x \right)}}{3 \left(x + 1\right) \sin{\left(x \right)}} - \frac{4 \left(x \left(x + 1\right) + \frac{3 x \left(x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 1\right)}{3 \left(x + 1\right)} - \frac{2 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right)}{3 \left(x + 1\right)^{2}} + \frac{4 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right) \cos{\left(x \right)}}{3 x \sin{\left(x \right)}} - \frac{4 \left(x \left(x + 1\right) + \frac{3 x \left(x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 1\right)}{3 x} - \frac{2 \left(- \frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x + 1\right) \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right) \cos{\left(x \right)}}{3 x \left(x + 1\right) \sin{\left(x \right)}} + \frac{4 \left(- \frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x + 1\right) \left(x \left(x + 1\right) + \frac{3 x \left(x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 1\right)}{9 x \left(x + 1\right)} - \frac{2 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right) \left(x \left(x + 1\right) + \frac{3 x \left(x + 1\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{2 \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 1\right)}{9 x \left(x + 1\right)} - \frac{2 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right)}{3 x \left(x + 1\right)} + \frac{\left(- \frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x + 1\right) \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right)}{3 x \left(x + 1\right)^{2}} - \frac{2 \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right)}{3 x^{2}} + \frac{\left(- \frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x + 1\right) \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right)}{3 x^{2} \left(x + 1\right)} - \frac{\left(- \frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x + 1\right)^{2} \left(\frac{2 x \left(x + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 x - 1\right)}{27 x^{2} \left(x + 1\right)^{2}}\right)}{x \left(x + 1\right)}$$

Simplificar

Gráfico

Otras calculadoras

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de ((x+1)*x/sin(x)^2)^(1/3)

Gráfico:

Definida a trozos:

Solución