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

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

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

The solution

You have entered [src]

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

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

x^4/(x^3 - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{4}$ and $g{\left(x \right)} = x^{3} - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

    /                 /         3          6   \                       \
    |               3 |     18*x       27*x    |         /          3 \|
    |              x *|1 - ------- + ----------|       3 |       3*x  ||
    |                 |          3            2|   12*x *|-1 + -------||
    |         3       |    -1 + x    /      3\ |         |           3||
    |     18*x        \              \-1 + x / /         \     -1 + x /|
6*x*|4 - ------- - ----------------------------- + --------------------|
    |          3                    3                          3       |
    \    -1 + x               -1 + x                     -1 + x        /
------------------------------------------------------------------------
                                      3                                 
                                -1 + x

$$\frac{6 x \left(\frac{12 x^{3} \left(\frac{3 x^{3}}{x^{3} - 1} - 1\right)}{x^{3} - 1} - \frac{x^{3} \left(\frac{27 x^{6}}{\left(x^{3} - 1\right)^{2}} - \frac{18 x^{3}}{x^{3} - 1} + 1\right)}{x^{3} - 1} - \frac{18 x^{3}}{x^{3} - 1} + 4\right)}{x^{3} - 1}$$

Simplify

The graph

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

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The solution