Derivada de x/((x+1)(x-2))

Ha introducido [src]

       x       
---------------
(x + 1)*(x - 2)

\frac{x}{\left(x - 2\right) \left(x + 1\right)}

x/(((x + 1)*(x - 2)))

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x$ y $g{\left(x \right)} = \left(x - 2\right) \left(x + 1\right)$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x + 1$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  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: $1$
  $g{\left(x \right)} = x - 2$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. diferenciamos $x - 2$ miembro por miembro:
    1. La derivada de una constante $-2$ es igual a cero.
    2. Según el principio, aplicamos: $x$ tenemos $1$
    Como resultado de: $1$
  Como resultado de: $2 x - 1$
Ahora aplicamos la regla de la derivada de una divesión:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

       1             x*(1 - 2*x)   
--------------- + -----------------
(x + 1)*(x - 2)          2        2
                  (x + 1) *(x - 2)

\frac{x \left(1 - 2 x\right)}{\left(x - 2\right)^{2} \left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)}

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Segunda derivada [src]

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

\frac{x \left(\left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right) - 2 + \frac{2 x - 1}{x + 1} + \frac{2 x - 1}{x - 2}\right) - 4 x + 2}{\left(x - 2\right)^{2} \left(x + 1\right)^{2}}

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Tercera derivada [src]

       /                                                                                                                     /  1       1   \              /  1       1   \                   \                                                              
       |                                                                                                          (-1 + 2*x)*|----- + ------|   (-1 + 2*x)*|----- + ------|                   |                                                              
       |    8       8                   /   1           1              1        \   3*(-1 + 2*x)   3*(-1 + 2*x)              \1 + x   -2 + x/              \1 + x   -2 + x/     4*(-1 + 2*x)  |   3*(-1 + 2*x)   3*(-1 + 2*x)                /  1       1   \
-6 - x*|- ----- - ------ + 2*(-1 + 2*x)*|-------- + --------- + ----------------| + ------------ + ------------ + --------------------------- + --------------------------- + ----------------| + ------------ + ------------ + 3*(-1 + 2*x)*|----- + ------|
       |  1 + x   -2 + x                |       2           2   (1 + x)*(-2 + x)|            2              2                1 + x                         -2 + x             (1 + x)*(-2 + x)|      1 + x          -2 + x                   \1 + x   -2 + x/
       \                                \(1 + x)    (-2 + x)                    /     (1 + x)       (-2 + x)                                                                                  /                                                              
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                             2         2                                                                                                                     
                                                                                                                      (1 + x) *(-2 + x)

\frac{- x \left(2 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right) + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{x + 1} - \frac{8}{x + 1} + \frac{3 \left(2 x - 1\right)}{\left(x + 1\right)^{2}} + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{x - 2} - \frac{8}{x - 2} + \frac{4 \left(2 x - 1\right)}{\left(x - 2\right) \left(x + 1\right)} + \frac{3 \left(2 x - 1\right)}{\left(x - 2\right)^{2}}\right) + 3 \left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right) - 6 + \frac{3 \left(2 x - 1\right)}{x + 1} + \frac{3 \left(2 x - 1\right)}{x - 2}}{\left(x - 2\right)^{2} \left(x + 1\right)^{2}}

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5-я производная [src]

  /                                                                                            /                                                                   /   1           1              1        \     /   1           1              1        \                                                                                                                                                                                                                                                                              /   1           1              1        \                /   1           1              1        \                /   1           1               1                   1        \                /   1           1               1                   1        \                                                                                                                                         /   1           1              1        \\                                                                                                                                                                                                /   1           1              1        \                 /   1           1              1        \                                                                        \
  |                                               /  1       1   \      /  1       1   \       |                                                                 4*|-------- + --------- + ----------------|   4*|-------- + --------- + ----------------|     /  1       1   \     /  1       1   \                                                                                                                                                 /  1       1   \              /  1       1   \     /  1       1   \   2*(-1 + 2*x)*|-------- + --------- + ----------------|   2*(-1 + 2*x)*|-------- + --------- + ----------------|   3*(-1 + 2*x)*|-------- + --------- + ----------------- + -----------------|   3*(-1 + 2*x)*|-------- + --------- + ----------------- + -----------------|                                                                           /  1       1   \              /  1       1   \   2*(-1 + 2*x)*|-------- + --------- + ----------------||                                                                                                                               /  1       1   \                /  1       1   \   10*(-1 + 2*x)*|-------- + --------- + ----------------|   10*(-1 + 2*x)*|-------- + --------- + ----------------|                                                        /  1       1   \|
  |                                            10*|----- + ------|   10*|----- + ------|       |                                                                   |       2           2   (1 + x)*(-2 + x)|     |       2           2   (1 + x)*(-2 + x)|   2*|----- + ------|   2*|----- + ------|                                                                                                                                      (-1 + 2*x)*|----- + ------|   (-1 + 2*x)*|----- + ------|   2*|----- + ------|                |       2           2   (1 + x)*(-2 + x)|                |       2           2   (1 + x)*(-2 + x)|                |       3           3                   2          2         |                |       3           3                   2          2         |                                                                (-1 + 2*x)*|----- + ------|   (-1 + 2*x)*|----- + ------|                |       2           2   (1 + x)*(-2 + x)||                                                                                                                  5*(-1 + 2*x)*|----- + ------|   5*(-1 + 2*x)*|----- + ------|                 |       2           2   (1 + x)*(-2 + x)|                 |       2           2   (1 + x)*(-2 + x)|                                           5*(-1 + 2*x)*|----- + ------||
  |     80          80            100             \1 + x   -2 + x/      \1 + x   -2 + x/       |     26          26              36                  36            \(1 + x)    (-2 + x)                    /     \(1 + x)    (-2 + x)                    /     \1 + x   -2 + x/     \1 + x   -2 + x/                /   1           1               1                   1                   1         \   10*(-1 + 2*x)   10*(-1 + 2*x)              \1 + x   -2 + x/              \1 + x   -2 + x/     \1 + x   -2 + x/                \(1 + x)    (-2 + x)                    /                \(1 + x)    (-2 + x)                    /                \(1 + x)    (-2 + x)    (1 + x)*(-2 + x)    (1 + x) *(-2 + x)/                \(1 + x)    (-2 + x)    (1 + x)*(-2 + x)    (1 + x) *(-2 + x)/     16*(-1 + 2*x)       16*(-1 + 2*x)       18*(-1 + 2*x)                 \1 + x   -2 + x/              \1 + x   -2 + x/                \(1 + x)    (-2 + x)                    /|                 /   1           1               1                   1        \   30*(-1 + 2*x)   30*(-1 + 2*x)                \1 + x   -2 + x/                \1 + x   -2 + x/                 \(1 + x)    (-2 + x)                    /                 \(1 + x)    (-2 + x)                    /     45*(-1 + 2*x)       45*(-1 + 2*x)                  \1 + x   -2 + x/|
2*|- -------- - --------- - ---------------- - ------------------- - ------------------- - 3*x*|- -------- - --------- - ----------------- - ----------------- - ------------------------------------------- - ------------------------------------------- - ------------------ - ------------------ + 4*(-1 + 2*x)*|-------- + --------- + ----------------- + ----------------- + ------------------| + ------------- + ------------- + --------------------------- + --------------------------- - ------------------ + ------------------------------------------------------ + ------------------------------------------------------ + --------------------------------------------------------------------------- + --------------------------------------------------------------------------- + ----------------- + ----------------- + ------------------ + --------------------------- + --------------------------- + ------------------------------------------------------| + 15*(-1 + 2*x)*|-------- + --------- + ----------------- + -----------------| + ------------- + ------------- + ----------------------------- + ----------------------------- + ------------------------------------------------------- + ------------------------------------------------------- + ----------------- + ----------------- + -----------------------------|
  |         2           2   (1 + x)*(-2 + x)          1 + x                 -2 + x             |         3           3                   2          2                               1 + x                                         -2 + x                                 2                    2                     |       4           4                   3          3                   2         2|             4               4                      3                             3             (1 + x)*(-2 + x)                                  2                                                        2                                                             1 + x                                                                         -2 + x                                                     3          3                   2         2                        2                    2                                    (1 + x)*(-2 + x)                   |                 |       3           3                   2          2         |             3               3                       2                               2                                      1 + x                                                     -2 + x                                           2          2                   (1 + x)*(-2 + x)      |
  \  (1 + x)    (-2 + x)                                                                       \  (1 + x)    (-2 + x)    (1 + x)*(-2 + x)    (1 + x) *(-2 + x)                                                                                                    (1 + x)             (-2 + x)                      \(1 + x)    (-2 + x)    (1 + x)*(-2 + x)    (1 + x) *(-2 + x)   (1 + x) *(-2 + x) /      (1 + x)        (-2 + x)                (1 + x)                      (-2 + x)                                                         (1 + x)                                                 (-2 + x)                                                                                                                                                                                       (1 + x)*(-2 + x)    (1 + x) *(-2 + x)   (1 + x) *(-2 + x)         (1 + x)*(-2 + x)              (1 + x) *(-2 + x)                                                              /                 \(1 + x)    (-2 + x)    (1 + x)*(-2 + x)    (1 + x) *(-2 + x)/      (1 + x)        (-2 + x)                 (1 + x)                        (-2 + x)                                                                                                                                  (1 + x)*(-2 + x)    (1 + x) *(-2 + x)                                /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 2         2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (1 + x) *(-2 + x)

\frac{2 \left(- 3 x \left(4 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{4}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)^{3}} + \frac{1}{\left(x - 2\right)^{2} \left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right)^{3} \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{4}}\right) + \frac{3 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{3}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right)^{2} \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{3}}\right)}{x + 1} - \frac{4 \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right)}{x + 1} + \frac{2 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right)}{\left(x + 1\right)^{2}} - \frac{2 \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x + 1\right)^{2}} + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x + 1\right)^{3}} - \frac{26}{\left(x + 1\right)^{3}} + \frac{10 \left(2 x - 1\right)}{\left(x + 1\right)^{4}} + \frac{3 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{3}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right)^{2} \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{3}}\right)}{x - 2} - \frac{4 \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right)}{x - 2} + \frac{2 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right) \left(x + 1\right)} - \frac{2 \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x - 2\right) \left(x + 1\right)} + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x - 2\right) \left(x + 1\right)^{2}} - \frac{36}{\left(x - 2\right) \left(x + 1\right)^{2}} + \frac{16 \left(2 x - 1\right)}{\left(x - 2\right) \left(x + 1\right)^{3}} + \frac{2 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right)^{2}} - \frac{2 \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x - 2\right)^{2}} + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x - 2\right)^{2} \left(x + 1\right)} - \frac{36}{\left(x - 2\right)^{2} \left(x + 1\right)} + \frac{18 \left(2 x - 1\right)}{\left(x - 2\right)^{2} \left(x + 1\right)^{2}} + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x - 2\right)^{3}} - \frac{26}{\left(x - 2\right)^{3}} + \frac{16 \left(2 x - 1\right)}{\left(x - 2\right)^{3} \left(x + 1\right)} + \frac{10 \left(2 x - 1\right)}{\left(x - 2\right)^{4}}\right) + 15 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{3}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right)^{2} \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{3}}\right) + \frac{10 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right)}{x + 1} - \frac{10 \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{x + 1} + \frac{5 \left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x + 1\right)^{2}} - \frac{80}{\left(x + 1\right)^{2}} + \frac{30 \left(2 x - 1\right)}{\left(x + 1\right)^{3}} + \frac{10 \left(2 x - 1\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 2\right) \left(x + 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right)}{x - 2} - \frac{10 \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{x - 2} + \frac{5 \left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x - 2\right) \left(x + 1\right)} - \frac{100}{\left(x - 2\right) \left(x + 1\right)} + \frac{45 \left(2 x - 1\right)}{\left(x - 2\right) \left(x + 1\right)^{2}} + \frac{5 \left(2 x - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 2}\right)}{\left(x - 2\right)^{2}} - \frac{80}{\left(x - 2\right)^{2}} + \frac{45 \left(2 x - 1\right)}{\left(x - 2\right)^{2} \left(x + 1\right)} + \frac{30 \left(2 x - 1\right)}{\left(x - 2\right)^{3}}\right)}{\left(x - 2\right)^{2} \left(x + 1\right)^{2}}

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Derivada de x/((x+1)(x-2))

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