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x^3/(x^2-4)
The derivative of the function/ x^3/(x^2-4)

Derivative of x^3/(x^2-4)

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

You have entered [src] 
   3  
  x   
------
 2    
x  - 4
x3x2−4\frac{x^{3}}{x^{2} - 4} 
x^3/(x^2 - 4)
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)=x3f{\left(x \right)} = x^{3} and g(x)=x2−4g{\left(x \right)} = x^{2} - 4.

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

    1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

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

    1. Differentiate x2−4x^{2} - 4 term by term:

      1. The derivative of the constant −4-4 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:

    −2x4+3x2(x2−4)(x2−4)2\frac{- 2 x^{4} + 3 x^{2} \left(x^{2} - 4\right)}{\left(x^{2} - 4\right)^{2}}

  2. Now simplify:

    x2(x2−12)(x2−4)2\frac{x^{2} \left(x^{2} - 12\right)}{\left(x^{2} - 4\right)^{2}}

The answer is:

x2(x2−12)(x2−4)2\frac{x^{2} \left(x^{2} - 12\right)}{\left(x^{2} - 4\right)^{2}}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        4         2 
     2*x       3*x  
- --------- + ------
          2    2    
  / 2    \    x  - 4
  \x  - 4/
−2x4(x2−4)2+3x2x2−4- \frac{2 x^{4}}{\left(x^{2} - 4\right)^{2}} + \frac{3 x^{2}}{x^{2} - 4}
Simplify
The second derivative [src] 
    /                 /          2 \\
    |               2 |       4*x  ||
    |              x *|-1 + -------||
    |         2       |           2||
    |      6*x        \     -4 + x /|
2*x*|3 - ------- + -----------------|
    |          2              2     |
    \    -4 + x         -4 + x      /
-------------------------------------
                     2               
               -4 + x
2x(x2(4x2x2−4−1)x2−4−6x2x2−4+3)x2−4\frac{2 x \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right)}{x^{2} - 4} - \frac{6 x^{2}}{x^{2} - 4} + 3\right)}{x^{2} - 4}
Simplify
The third derivative [src] 
  /                   /          2 \        /          2 \\
  |                 4 |       2*x  |      2 |       4*x  ||
  |              4*x *|-1 + -------|   3*x *|-1 + -------||
  |         2         |           2|        |           2||
  |      6*x          \     -4 + x /        \     -4 + x /|
6*|1 - ------- - ------------------- + -------------------|
  |          2                 2                   2      |
  |    -4 + x         /      2\              -4 + x       |
  \                   \-4 + x /                           /
-----------------------------------------------------------
                                2                          
                          -4 + x
6(−4x4(2x2x2−4−1)(x2−4)2+3x2(4x2x2−4−1)x2−4−6x2x2−4+1)x2−4\frac{6 \left(- \frac{4 x^{4} \left(\frac{2 x^{2}}{x^{2} - 4} - 1\right)}{\left(x^{2} - 4\right)^{2}} + \frac{3 x^{2} \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right)}{x^{2} - 4} - \frac{6 x^{2}}{x^{2} - 4} + 1\right)}{x^{2} - 4}
Simplify
The graph
Derivative of x^3/(x^2-4)
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