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

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

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   4  
  x   
------
 3    
x  - 1
x4x3−1\frac{x^{4}}{x^{3} - 1} 
x^4/(x^3 - 1)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x4f{\left(x \right)} = x^{4} and g(x)=x3−1g{\left(x \right)} = x^{3} - 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x4x^{4} goes to 4x34 x^{3}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x3−1x^{3} - 1 term by term:

      1. The derivative of the constant −1-1 is zero.

      2. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      The result is: 3x23 x^{2}

    Now plug in to the quotient rule:

    −3x6+4x3(x3−1)(x3−1)2\frac{- 3 x^{6} + 4 x^{3} \left(x^{3} - 1\right)}{\left(x^{3} - 1\right)^{2}}

  2. Now simplify:

    x3(x3−4)(x3−1)2\frac{x^{3} \left(x^{3} - 4\right)}{\left(x^{3} - 1\right)^{2}}

The answer is:

x3(x3−4)(x3−1)2\frac{x^{3} \left(x^{3} - 4\right)}{\left(x^{3} - 1\right)^{2}}

The graph
02468-8-6-4-2-1010-5050
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/
−3x6(x3−1)2+4x3x3−1- \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
6x2(x3(3x3x3−1−1)x3−1−4x3x3−1+2)x3−1\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
6x(12x3(3x3x3−1−1)x3−1−x3(27x6(x3−1)2−18x3x3−1+1)x3−1−18x3x3−1+4)x3−1\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
Derivative of x^4/(x^3-1)
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