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Derivative of (x-2)^4
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Graphing y =:
(x^2-1)/(x^3+1)
How do you in partial fractions?:
(x^2-1)/(x^3+1)
Identical expressions
(x^ two - one)/(x^ three + one)
(x squared minus 1) divide by (x cubed plus 1)
(x to the power of two minus one) divide by (x to the power of three plus 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:

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

The answer is:

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

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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Identical expressions

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

The graph:

Piecewise:

The solution