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Graphing y =:
(x+3)/(x^2-9)
Identical expressions
(x+ three)/(x^ two - nine)
(x plus 3) divide by (x squared minus 9)
(x plus three) divide by (x to the power of two minus nine)
(x+3)/(x2-9)
x+3/x2-9
(x+3)/(x²-9)
(x+3)/(x to the power of 2-9)
x+3/x^2-9
(x+3) divide by (x^2-9)
Similar expressions
(x+3)/(x^2+9)
(x-3)/(x^2-9)

The derivative of the function/ (x+3)/(x^2-9)

Derivative of (x+3)/(x^2-9)

The solution

You have entered [src]

x + 3 
------
 2    
x  - 9

$$\frac{x + 3}{x^{2} - 9}$$

(x + 3)/(x^2 - 9)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x + 3$ and $g{\left(x \right)} = x^{2} - 9$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x + 3$ term by term:
  1. The derivative of the constant $3$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} - 9$ term by term:
  1. The derivative of the constant $-9$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

$\frac{x^{2} - 2 x \left(x + 3\right) - 9}{\left(x^{2} - 9\right)^{2}}$
Now simplify:

$- \frac{1}{x^{2} - 6 x + 9}$

The answer is:

$- \frac{1}{x^{2} - 6 x + 9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1      2*x*(x + 3)
------ - -----------
 2                2 
x  - 9    / 2    \  
          \x  - 9/

$$- \frac{2 x \left(x + 3\right)}{\left(x^{2} - 9\right)^{2}} + \frac{1}{x^{2} - 9}$$

Simplify

The second derivative [src]

  /       /          2 \        \
  |       |       4*x  |        |
2*|-2*x + |-1 + -------|*(3 + x)|
  |       |           2|        |
  \       \     -9 + x /        /
---------------------------------
                     2           
            /      2\            
            \-9 + x /

$$\frac{2 \left(- 2 x + \left(x + 3\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right)\right)}{\left(x^{2} - 9\right)^{2}}$$

Simplify

The third derivative [src]

  /                   /          2 \        \
  |                   |       2*x  |        |
  |               4*x*|-1 + -------|*(3 + x)|
  |          2        |           2|        |
  |       4*x         \     -9 + x /        |
6*|-1 + ------- - --------------------------|
  |           2                  2          |
  \     -9 + x             -9 + x           /
---------------------------------------------
                           2                 
                  /      2\                  
                  \-9 + x /

$$\frac{6 \left(\frac{4 x^{2}}{x^{2} - 9} - \frac{4 x \left(x + 3\right) \left(\frac{2 x^{2}}{x^{2} - 9} - 1\right)}{x^{2} - 9} - 1\right)}{\left(x^{2} - 9\right)^{2}}$$

Simplify

The graph

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Derivative of (x+3)/(x^2-9)

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The solution