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

Function f() - derivative -N order at the point
v

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The solution

You have entered [src]
     1      
------------
 2          
x  + 3*x + 2
1(x2+3x)+2\frac{1}{\left(x^{2} + 3 x\right) + 2}
1/(x^2 + 3*x + 2)
Detail solution
  1. Let u=(x2+3x)+2u = \left(x^{2} + 3 x\right) + 2.

  2. Apply the power rule: 1u\frac{1}{u} goes to 1u2- \frac{1}{u^{2}}

  3. Then, apply the chain rule. Multiply by ddx((x2+3x)+2)\frac{d}{d x} \left(\left(x^{2} + 3 x\right) + 2\right):

    1. Differentiate (x2+3x)+2\left(x^{2} + 3 x\right) + 2 term by term:

      1. Differentiate x2+3xx^{2} + 3 x term by term:

        1. Apply the power rule: x2x^{2} goes to 2x2 x

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: xx goes to 11

          So, the result is: 33

        The result is: 2x+32 x + 3

      2. The derivative of the constant 22 is zero.

      The result is: 2x+32 x + 3

    The result of the chain rule is:

    2x+3((x2+3x)+2)2- \frac{2 x + 3}{\left(\left(x^{2} + 3 x\right) + 2\right)^{2}}

  4. Now simplify:

    2x+3(x2+3x+2)2- \frac{2 x + 3}{\left(x^{2} + 3 x + 2\right)^{2}}


The answer is:

2x+3(x2+3x+2)2- \frac{2 x + 3}{\left(x^{2} + 3 x + 2\right)^{2}}

The graph
02468-8-6-4-2-1010-250250
The first derivative [src]
    -3 - 2*x   
---------------
              2
/ 2          \ 
\x  + 3*x + 2/ 
2x3((x2+3x)+2)2\frac{- 2 x - 3}{\left(\left(x^{2} + 3 x\right) + 2\right)^{2}}
The second derivative [src]
  /               2 \
  |      (3 + 2*x)  |
2*|-1 + ------------|
  |          2      |
  \     2 + x  + 3*x/
---------------------
                 2   
   /     2      \    
   \2 + x  + 3*x/    
2((2x+3)2x2+3x+21)(x2+3x+2)2\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}}
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/             
24((2x+3)4(x2+3x+2)23(2x+3)2x2+3x+2+1)(x2+3x+2)3\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}}
The third derivative [src]
  /              2 \          
  |     (3 + 2*x)  |          
6*|2 - ------------|*(3 + 2*x)
  |         2      |          
  \    2 + x  + 3*x/          
------------------------------
                     3        
       /     2      \         
       \2 + x  + 3*x/         
6(2x+3)((2x+3)2x2+3x+2+2)(x2+3x+2)3\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}}