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Integral of d{x}:
x^3/(x^4-1)
How do you in partial fractions?:
x^3/(x^4-1)
Graphing y =:
x^3/(x^4-1)
Identical expressions
x^ three /(x^ four - one)
x cubed divide by (x to the power of 4 minus 1)
x to the power of three divide by (x to the power of four minus one)
x3/(x4-1)
x3/x4-1
x³/(x⁴-1)
x to the power of 3/(x to the power of 4-1)
x^3/x^4-1
x^3 divide by (x^4-1)
Similar expressions
x^3/(x^4+1)

The derivative of the function/ x^3/(x^4-1)

Derivative of x^3/(x^4-1)

The solution

You have entered [src]

   3  
  x   
------
 4    
x  - 1

$$\frac{x^{3}}{x^{4} - 1}$$

x^3/(x^4 - 1)

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 $g{\left(x \right)} = x^{4} - 1$ .

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 $x^{4} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  The result is: $4 x^{3}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

    /                   /          4 \\
    |                 4 |       8*x  ||
    |              2*x *|-3 + -------||
    |         4         |           4||
    |     12*x          \     -1 + x /|
2*x*|3 - ------- + -------------------|
    |          4               4      |
    \    -1 + x          -1 + x       /
---------------------------------------
                      4                
                -1 + x

$$\frac{2 x \left(\frac{2 x^{4} \left(\frac{8 x^{4}}{x^{4} - 1} - 3\right)}{x^{4} - 1} - \frac{12 x^{4}}{x^{4} - 1} + 3\right)}{x^{4} - 1}$$

Simplify

The third derivative [src]

  /                   /         4          8   \                      \
  |                 4 |     12*x       16*x    |        /          4 \|
  |              4*x *|1 - ------- + ----------|      4 |       8*x  ||
  |                   |          4            2|   6*x *|-3 + -------||
  |         4         |    -1 + x    /      4\ |        |           4||
  |     12*x          \              \-1 + x / /        \     -1 + x /|
6*|1 - ------- - ------------------------------- + -------------------|
  |          4                     4                           4      |
  \    -1 + x                -1 + x                      -1 + x       /
-----------------------------------------------------------------------
                                      4                                
                                -1 + x

$$\frac{6 \left(\frac{6 x^{4} \left(\frac{8 x^{4}}{x^{4} - 1} - 3\right)}{x^{4} - 1} - \frac{4 x^{4} \left(\frac{16 x^{8}}{\left(x^{4} - 1\right)^{2}} - \frac{12 x^{4}}{x^{4} - 1} + 1\right)}{x^{4} - 1} - \frac{12 x^{4}}{x^{4} - 1} + 1\right)}{x^{4} - 1}$$

Simplify

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Derivative of x^3/(x^4-1)

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The solution