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

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
x + 2 
------
 2    
x  - 9
x+2x2−9\frac{x + 2}{x^{2} - 9} 
(x + 2)/(x^2 - 9)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x+2f{\left(x \right)} = x + 2 and g(x)=x2−9g{\left(x \right)} = x^{2} - 9.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate x+2x + 2 term by term:

      1. The derivative of the constant 22 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x2−9x^{2} - 9 term by term:

      1. The derivative of the constant −9-9 is zero.

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

      The result is: 2x2 x

    Now plug in to the quotient rule:

    x2−2x(x+2)−9(x2−9)2\frac{x^{2} - 2 x \left(x + 2\right) - 9}{\left(x^{2} - 9\right)^{2}}

The answer is:

x2−2x(x+2)−9(x2−9)2\frac{x^{2} - 2 x \left(x + 2\right) - 9}{\left(x^{2} - 9\right)^{2}}

The graph
02468-8-6-4-2-1010-200200
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  1      2*x*(x + 2)
------ - -----------
 2                2 
x  - 9    / 2    \  
          \x  - 9/
−2x(x+2)(x2−9)2+1x2−9- \frac{2 x \left(x + 2\right)}{\left(x^{2} - 9\right)^{2}} + \frac{1}{x^{2} - 9}
Simplify
The second derivative [src] 
  /       /          2 \        \
  |       |       4*x  |        |
2*|-2*x + |-1 + -------|*(2 + x)|
  |       |           2|        |
  \       \     -9 + x /        /
---------------------------------
                     2           
            /      2\            
            \-9 + x /
2(−2x+(x+2)(4x2x2−9−1))(x2−9)2\frac{2 \left(- 2 x + \left(x + 2\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 + -------|*(2 + x)|
  |          2        |           2|        |
  |       4*x         \     -9 + x /        |
6*|-1 + ------- - --------------------------|
  |           2                  2          |
  \     -9 + x             -9 + x           /
---------------------------------------------
                           2                 
                  /      2\                  
                  \-9 + x /
6(4x2x2−9−4x(x+2)(2x2x2−9−1)x2−9−1)(x2−9)2\frac{6 \left(\frac{4 x^{2}}{x^{2} - 9} - \frac{4 x \left(x + 2\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
Derivative of (x+2)/(x^2-9)
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