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

Ha introducido [src]

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

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

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

Solución detallada

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

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

$\frac{\left(x + 2\right) \left(3 x + 4\right)}{3 \left(x + 1\right)^{\frac{2}{3}} \left|{\left(x + 2\right)^{\frac{4}{3}}}\right|}$

Respuesta:

$\frac{\left(x + 2\right) \left(3 x + 4\right)}{3 \left(x + 1\right)^{\frac{2}{3}} \left|{\left(x + 2\right)^{\frac{4}{3}}}\right|}$

Primera derivada [src]

                     /       2                    \
3 _______        2/3 |(x + 2)    (4 + 2*x)*(x + 1)|
\/ x + 1 *|x + 2|   *|-------- + -----------------|
                     \   3               3        /
---------------------------------------------------
                                 2                 
                  (x + 1)*(x + 2)

\frac{\sqrt[3]{x + 1} \left|{x + 2}\right|^{\frac{2}{3}} \left(\frac{\left(x + 1\right) \left(2 x + 4\right)}{3} + \frac{\left(x + 2\right)^{2}}{3}\right)}{\left(x + 1\right) \left(x + 2\right)^{2}}

Simplificar

Segunda derivada [src]

          /       2/3     3 _______            \                                                                           
          ||2 + x|      2*\/ 1 + x *sign(2 + x)|                                                                           
(4 + 3*x)*|---------- + -----------------------|                                                                           
          |       2/3         3 _________      |            2/3                      2/3                      2/3          
          \(1 + x)            \/ |2 + x|       /   3*|2 + x|   *(4 + 3*x)   6*|2 + x|   *(4 + 3*x)   6*|2 + x|   *(5 + 3*x)
------------------------------------------------ - ---------------------- - ---------------------- + ----------------------
                     1 + x                                      5/3                  2/3                      2/3          
                                                         (1 + x)              (1 + x)   *(2 + x)       (1 + x)   *(2 + x)  
---------------------------------------------------------------------------------------------------------------------------
                                                         9*(2 + x)

\frac{\frac{\left(3 x + 4\right) \left(\frac{2 \sqrt[3]{x + 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}}}\right)}{x + 1} - \frac{6 \left(3 x + 4\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}} \left(x + 2\right)} + \frac{6 \left(3 x + 5\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}} \left(x + 2\right)} - \frac{3 \left(3 x + 4\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{5}{3}}}}{9 \left(x + 2\right)}

Simplificar

Tercera derivada [src]

                               /       2/3     3 _______            \               /         2/3   3 _______     2                                     3 _______                  \                                                                                                                    /       2/3     3 _______            \                                          /       2/3     3 _______            \                                                                                                
                               ||2 + x|      2*\/ 1 + x *sign(2 + x)|               |  |2 + x|      \/ 1 + x *sign (2 + x)       2*sign(2 + x)        6*\/ 1 + x *DiracDelta(2 + x)|                                                                                                                    ||2 + x|      2*\/ 1 + x *sign(2 + x)|                                          ||2 + x|      2*\/ 1 + x *sign(2 + x)|                                                                                                
                     (4 + 3*x)*|---------- + -----------------------|   2*(4 + 3*x)*|- ---------- - ---------------------- + ---------------------- + -----------------------------|                                                                                                        2*(4 + 3*x)*|---------- + -----------------------|                              2*(5 + 3*x)*|---------- + -----------------------|                                                                                                
            2/3                |       2/3         3 _________      |               |         5/3                4/3                2/3 3 _________            3 _________         |            2/3                      2/3                       2/3                      2/3                         |       2/3         3 _________      |                                          |       2/3         3 _________      |             2/3                                                                                
   2*|2 + x|                   \(1 + x)            \/ |2 + x|       /               \  (1 + x)            |2 + x|            (1 + x)   *\/ |2 + x|             \/ |2 + x|          /   5*|2 + x|   *(4 + 3*x)   2*|2 + x|   *(4 + 3*x)   10*|2 + x|   *(5 + 3*x)   8*|2 + x|   *(5 + 3*x)               \(1 + x)            \/ |2 + x|       /   2*(4 + 3*x)*sign(2 + x)                \(1 + x)            \/ |2 + x|       /   10*|2 + x|   *(4 + 3*x)       4*(4 + 3*x)*sign(2 + x)            4*(5 + 3*x)*sign(2 + x)     
------------------ - ------------------------------------------------ + ------------------------------------------------------------------------------------------------------------ + ---------------------- + ---------------------- - ----------------------- - ---------------------- - -------------------------------------------------- - ------------------------ + -------------------------------------------------- + ----------------------- - -------------------------------- + --------------------------------
       2/3                                       2                                                                       27*(1 + x)                                                                  8/3                2/3        2                5/3                     2/3        2                    9*(1 + x)*(2 + x)                             5/3 3 _________                   9*(1 + x)*(2 + x)                               5/3                     2/3         3 _________            2/3         3 _________
(1 + x)   *(2 + x)                      9*(1 + x)                                                                                                                                           9*(1 + x)            (1 + x)   *(2 + x)        9*(1 + x)   *(2 + x)    3*(1 + x)   *(2 + x)                                                          9*(1 + x)   *\/ |2 + x|                                                           9*(1 + x)   *(2 + x)    9*(1 + x)   *(2 + x)*\/ |2 + x|    9*(1 + x)   *(2 + x)*\/ |2 + x| 
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                                                                                                                                                                                                                                                            2 + x

\frac{\frac{2 \left(3 x + 4\right) \left(\frac{6 \sqrt[3]{x + 1} \delta\left(x + 2\right)}{\sqrt[3]{\left|{x + 2}\right|}} - \frac{\sqrt[3]{x + 1} \operatorname{sign}^{2}{\left(x + 2 \right)}}{\left|{x + 2}\right|^{\frac{4}{3}}} + \frac{2 \operatorname{sign}{\left(x + 2 \right)}}{\left(x + 1\right)^{\frac{2}{3}} \sqrt[3]{\left|{x + 2}\right|}} - \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{5}{3}}}\right)}{27 \left(x + 1\right)} - \frac{2 \left(3 x + 4\right) \left(\frac{2 \sqrt[3]{x + 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}}}\right)}{9 \left(x + 1\right) \left(x + 2\right)} + \frac{2 \left(3 x + 5\right) \left(\frac{2 \sqrt[3]{x + 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}}}\right)}{9 \left(x + 1\right) \left(x + 2\right)} - \frac{\left(3 x + 4\right) \left(\frac{2 \sqrt[3]{x + 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}}}\right)}{9 \left(x + 1\right)^{2}} - \frac{4 \left(3 x + 4\right) \operatorname{sign}{\left(x + 2 \right)}}{9 \left(x + 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} + \frac{4 \left(3 x + 5\right) \operatorname{sign}{\left(x + 2 \right)}}{9 \left(x + 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} + \frac{2 \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}} \left(x + 2\right)} + \frac{2 \left(3 x + 4\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x + 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} - \frac{8 \left(3 x + 5\right) \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x + 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} - \frac{2 \left(3 x + 4\right) \operatorname{sign}{\left(x + 2 \right)}}{9 \left(x + 1\right)^{\frac{5}{3}} \sqrt[3]{\left|{x + 2}\right|}} + \frac{10 \left(3 x + 4\right) \left|{x + 2}\right|^{\frac{2}{3}}}{9 \left(x + 1\right)^{\frac{5}{3}} \left(x + 2\right)} - \frac{10 \left(3 x + 5\right) \left|{x + 2}\right|^{\frac{2}{3}}}{9 \left(x + 1\right)^{\frac{5}{3}} \left(x + 2\right)} + \frac{5 \left(3 x + 4\right) \left|{x + 2}\right|^{\frac{2}{3}}}{9 \left(x + 1\right)^{\frac{8}{3}}}}{x + 2}

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

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