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

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

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

The solution

You have entered [src]

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

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

x^2/(x^3 - 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     /            6           3 \
   2 |        27*x        36*x  |
6*x *|-10 - ---------- + -------|
     |               2         3|
     |      /      3\    -1 + x |
     \      \-1 + x /           /
---------------------------------
                     2           
            /      3\            
            \-1 + x /

$$\frac{6 x^{2} \left(- \frac{27 x^{6}}{\left(x^{3} - 1\right)^{2}} + \frac{36 x^{3}}{x^{3} - 1} - 10\right)}{\left(x^{3} - 1\right)^{2}}$$

Simplify

The graph

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

The graph:

Piecewise:

The solution