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(x^4-x^2)/(x-1)
Identical expressions
(x^ four -x^ two)/(x- one)
(x to the power of 4 minus x squared ) divide by (x minus 1)
(x to the power of four minus x to the power of two) divide by (x minus one)
(x4-x2)/(x-1)
x4-x2/x-1
(x⁴-x²)/(x-1)
(x to the power of 4-x to the power of 2)/(x-1)
x^4-x^2/x-1
(x^4-x^2) divide by (x-1)
Similar expressions
(x^4-x^2)/(x+1)
(x^4+x^2)/(x-1)

The derivative of the function/ (x^4-x^2)/(x-1)

Derivative of (x^4-x^2)/(x-1)

The solution

You have entered [src]

 4    2
x  - x 
-------
 x - 1

$$\frac{x^{4} - x^{2}}{x - 1}$$

(x^4 - x^2)/(x - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{4} - x^{2}$ and $g{\left(x \right)} = x - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{4} - x^{2}$ term by term:
  1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  The result is: $4 x^{3} - 2 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

$x \left(3 x + 2\right)$

The answer is:

$x \left(3 x + 2\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          3    4    2 
-2*x + 4*x    x  - x  
----------- - --------
   x - 1             2
              (x - 1)

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

Simplify

The second derivative [src]

  /             2 /      2\       /        2\\
  |        2   x *\-1 + x /   2*x*\-1 + 2*x /|
2*|-1 + 6*x  + ------------ - ---------------|
  |                     2          -1 + x    |
  \             (-1 + x)                     /
----------------------------------------------
                    -1 + x

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (x^4-x^2)/(x-1)

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The solution