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Derivative of:
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Derivative of -(x^2+324)/x
Graphing y =:
(x^3)/(4-x^2)
Integral of d{x}:
(x^3)/(4-x^2)
Identical expressions
(x^ three)/(four -x^ two)
(x cubed ) divide by (4 minus x squared )
(x to the power of three) divide by (four minus x to the power of two)
(x3)/(4-x2)
x3/4-x2
(x³)/(4-x²)
(x to the power of 3)/(4-x to the power of 2)
x^3/4-x^2
(x^3) divide by (4-x^2)
Similar expressions
(x^3)/(4+x^2)

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

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

The solution

You have entered [src]

   3  
  x   
------
     2
4 - x

$$\frac{x^{3}}{4 - x^{2}}$$

x^3/(4 - x^2)

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 $g{\left(x \right)} = 4 - x^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $4 - x^{2}$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  The result is: $- 2 x$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      4         2 
   2*x       3*x  
--------- + ------
        2        2
/     2\    4 - x 
\4 - x /

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

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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Piecewise:

The solution