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Integral of d{x}:
(x^2+1)/(x^3-1)
Graphing y =:
(x^2+1)/(x^3-1)
Identical expressions
(x^ two + one)/(x^ three - one)
(x squared plus 1) divide by (x cubed minus 1)
(x to the power of two plus one) divide by (x to the power of three minus one)
(x2+1)/(x3-1)
x2+1/x3-1
(x²+1)/(x³-1)
(x to the power of 2+1)/(x to the power of 3-1)
x^2+1/x^3-1
(x^2+1) divide by (x^3-1)
Similar expressions
(x^2-1)/(x^3-1)
(x^2+1)/(x^3+1)

The derivative of the function/ (x^2+1)/(x^3-1)

Derivative of (x^2+1)/(x^3-1)

The solution

You have entered [src]

 2    
x  + 1
------
 3    
x  - 1

$$\frac{x^{2} + 1}{x^{3} - 1}$$

(x^2 + 1)/(x^3 - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} + 1$ and $g{\left(x \right)} = x^{3} - 1$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                  /         3          6   \        /          3 \\
  |     2   /     2\ |     18*x       27*x    |      2 |       3*x  ||
6*|- 3*x  - \1 + x /*|1 - ------- + ----------| + 6*x *|-1 + -------||
  |                  |          3            2|        |           3||
  |                  |    -1 + x    /      3\ |        \     -1 + x /|
  \                  \              \-1 + x / /                      /
----------------------------------------------------------------------
                                       2                              
                              /      3\                               
                              \-1 + x /

$$\frac{6 \left(6 x^{2} \left(\frac{3 x^{3}}{x^{3} - 1} - 1\right) - 3 x^{2} - \left(x^{2} + 1\right) \left(\frac{27 x^{6}}{\left(x^{3} - 1\right)^{2}} - \frac{18 x^{3}}{x^{3} - 1} + 1\right)\right)}{\left(x^{3} - 1\right)^{2}}$$

Simplify

The graph

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Derivative of (x^2+1)/(x^3-1)

The graph:

Piecewise:

The solution