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(x^ two - three *x+ one)/(x^ two +x+ one)
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(x^2-3*x-1)/(x^2+x+1)
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(x^2-3*x+1)/(x^2-x+1)

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

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

The solution

You have entered [src]

 2          
x  - 3*x + 1
------------
  2         
 x  + x + 1

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

  / 2          \
d |x  - 3*x + 1|
--|------------|
dx|  2         |
  \ x  + x + 1 /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                        / 2          \
 -3 + 2*x    (-1 - 2*x)*\x  - 3*x + 1/
---------- + -------------------------
 2                             2      
x  + x + 1         / 2        \       
                   \x  + x + 1/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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Piecewise:

The solution