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Derivative of:
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Graphing y =:
1/(x^2+2x+3)
Integral of d{x}:
1/(x^2+2x+3)
Identical expressions
one /(x^ two +2x+ three)
1 divide by (x squared plus 2x plus 3)
one divide by (x to the power of two plus 2x plus three)
1/(x2+2x+3)
1/x2+2x+3
1/(x²+2x+3)
1/(x to the power of 2+2x+3)
1/x^2+2x+3
1 divide by (x^2+2x+3)
Similar expressions
1/(x^2-2x+3)
1/(x^2+2x-3)

The derivative of the function/ 1/(x^2+2x+3)

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

The solution

You have entered [src]

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

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

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

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

The graph:

Piecewise:

The solution