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Integral of d{x}:
1/(x^2+3x+2)
Identical expressions
one /(x^ two +3x+ two)
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Similar expressions
1/(x^2+3x-2)
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The derivative of the function/ 1/(x^2+3x+2)

Derivative of 1/(x^2+3x+2)

The solution

You have entered [src]

     1      
------------
 2          
x  + 3*x + 2

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

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

Detail solution

Let $u = \left(x^{2} + 3 x\right) + 2$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 3 x\right) + 2\right)$ :
1. Differentiate $\left(x^{2} + 3 x\right) + 2$ term by term:
  1. Differentiate $x^{2} + 3 x$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result is: $2 x + 3$
  2. The derivative of the constant $2$ is zero.
  The result is: $2 x + 3$
The result of the chain rule is:

$- \frac{2 x + 3}{\left(\left(x^{2} + 3 x\right) + 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    -3 - 2*x   
---------------
              2
/ 2          \ 
\x  + 3*x + 2/

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

Simplify

The second derivative [src]

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

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

Simplify

4-я производная [src]

   /                4                2\
   |       (3 + 2*x)      3*(3 + 2*x) |
24*|1 + --------------- - ------------|
   |                  2        2      |
   |    /     2      \    2 + x  + 3*x|
   \    \2 + x  + 3*x/                /
---------------------------------------
                          3            
            /     2      \             
            \2 + x  + 3*x/

$$\frac{24 \left(\frac{\left(2 x + 3\right)^{4}}{\left(x^{2} + 3 x + 2\right)^{2}} - \frac{3 \left(2 x + 3\right)^{2}}{x^{2} + 3 x + 2} + 1\right)}{\left(x^{2} + 3 x + 2\right)^{3}}$$

Simplify

The third derivative [src]

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

$$\frac{6 \left(2 x + 3\right) \left(- \frac{\left(2 x + 3\right)^{2}}{x^{2} + 3 x + 2} + 2\right)}{\left(x^{2} + 3 x + 2\right)^{3}}$$

Simplify

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Derivative of 1/(x^2+3x+2)

The graph:

Piecewise:

The solution