Derivative of 1/(x(x+2))

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

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

1/(x*(x + 2))

Detail solution

Let $u = x \left(x + 2\right)$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \left(x + 2\right)$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = x + 2$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x + 2$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $2$ is zero.
    The result is: $1$
  The result is: $2 x + 2$
The result of the chain rule is:

$- \frac{2 x + 2}{x^{2} \left(x + 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 1/(x(x+2))

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The solution