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Derivative of y=x^3-8x^2+10x-4
Derivative of x/(x^2+4)
Identical expressions
(5x^ two -2x+ four)/((x- one)*(x+ one))
(5x squared minus 2x plus 4) divide by ((x minus 1) multiply by (x plus 1))
(5x to the power of two minus 2x plus four) divide by ((x minus one) multiply by (x plus one))
(5x2-2x+4)/((x-1)*(x+1))
5x2-2x+4/x-1*x+1
(5x²-2x+4)/((x-1)*(x+1))
(5x to the power of 2-2x+4)/((x-1)*(x+1))
(5x^2-2x+4)/((x-1)(x+1))
(5x2-2x+4)/((x-1)(x+1))
5x2-2x+4/x-1x+1
5x^2-2x+4/x-1x+1
(5x^2-2x+4) divide by ((x-1)*(x+1))
Similar expressions
(5x^2-2x+4)/((x+1)*(x+1))
(5x^2-2x+4)/((x-1)*(x-1))
(5x^2+2x+4)/((x-1)*(x+1))
(5x^2-2x-4)/((x-1)*(x+1))

The derivative of the function/ (5x^2-2x+4)/((x-1)*(x+1))

Derivative of (5x^2-2x+4)/((x-1)*(x+1))

The solution

You have entered [src]

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

$$\frac{\left(5 x^{2} - 2 x\right) + 4}{\left(x - 1\right) \left(x + 1\right)}$$

(5*x^2 - 2*x + 4)/(((x - 1)*(x + 1)))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 5 x^{2} - 2 x + 4$ and $g{\left(x \right)} = \left(x - 1\right) \left(x + 1\right)$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $5 x^{2} - 2 x + 4$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. 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: $-2$
  3. 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: $10 x$
  The result is: $10 x - 2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x + 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  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: $1$
  $g{\left(x \right)} = x - 1$ ; 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: $1$
  The result is: $2 x$
Now plug in to the quotient rule:

$\frac{- 2 x \left(5 x^{2} - 2 x + 4\right) + \left(x - 1\right) \left(x + 1\right) \left(10 x - 2\right)}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}}$
Now simplify:

$\frac{2 \left(x^{2} - 9 x + 1\right)}{x^{4} - 2 x^{2} + 1}$

The answer is:

$\frac{2 \left(x^{2} - 9 x + 1\right)}{x^{4} - 2 x^{2} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                  /   2          \
       1                      2*x*\5*x  - 2*x + 4/
---------------*(-2 + 10*x) - --------------------
(x + 1)*(x - 1)                       2        2  
                               (x + 1) *(x - 1)

$$- \frac{2 x \left(\left(5 x^{2} - 2 x\right) + 4\right)}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}} + \frac{1}{\left(x - 1\right) \left(x + 1\right)} \left(10 x - 2\right)$$

Simplify

The second derivative [src]

  /    /             2\ /       x       x        /  1       1   \\                   \
  |    \4 - 2*x + 5*x /*|-1 + ----- + ------ + x*|----- + ------||                   |
  |                     \     1 + x   -1 + x     \1 + x   -1 + x//    4*x*(-1 + 5*x) |
2*|5 + ----------------------------------------------------------- - ----------------|
  \                          (1 + x)*(-1 + x)                        (1 + x)*(-1 + x)/
--------------------------------------------------------------------------------------
                                   (1 + x)*(-1 + x)

$$\frac{2 \left(- \frac{4 x \left(5 x - 1\right)}{\left(x - 1\right) \left(x + 1\right)} + 5 + \frac{\left(5 x^{2} - 2 x + 4\right) \left(x \left(\frac{1}{x + 1} + \frac{1}{x - 1}\right) + \frac{x}{x + 1} + \frac{x}{x - 1} - 1\right)}{\left(x - 1\right) \left(x + 1\right)}\right)}{\left(x - 1\right) \left(x + 1\right)}$$

Simplify

The third derivative [src]

  /                         /                                                                                            /  1       1   \     /  1       1   \                   \                                                          \
  |                         |                                                                                          x*|----- + ------|   x*|----- + ------|                   |                                                          |
  |        /             2\ |    4       4          /   1           1              1        \     3*x         3*x        \1 + x   -1 + x/     \1 + x   -1 + x/         4*x       |                /       x       x        /  1       1   \\|
2*|-30*x - \4 - 2*x + 5*x /*|- ----- - ------ + 2*x*|-------- + --------- + ----------------| + -------- + --------- + ------------------ + ------------------ + ----------------| + 6*(-1 + 5*x)*|-1 + ----- + ------ + x*|----- + ------|||
  |                         |  1 + x   -1 + x       |       2           2   (1 + x)*(-1 + x)|          2           2         1 + x                -1 + x         (1 + x)*(-1 + x)|                \     1 + x   -1 + x     \1 + x   -1 + x//|
  \                         \                       \(1 + x)    (-1 + x)                    /   (1 + x)    (-1 + x)                                                              /                                                          /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                     2         2                                                                                                             
                                                                                                              (1 + x) *(-1 + x)

$$\frac{2 \left(- 30 x + 6 \left(5 x - 1\right) \left(x \left(\frac{1}{x + 1} + \frac{1}{x - 1}\right) + \frac{x}{x + 1} + \frac{x}{x - 1} - 1\right) - \left(5 x^{2} - 2 x + 4\right) \left(2 x \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 1\right) \left(x + 1\right)} + \frac{1}{\left(x - 1\right)^{2}}\right) + \frac{x \left(\frac{1}{x + 1} + \frac{1}{x - 1}\right)}{x + 1} + \frac{3 x}{\left(x + 1\right)^{2}} + \frac{x \left(\frac{1}{x + 1} + \frac{1}{x - 1}\right)}{x - 1} + \frac{4 x}{\left(x - 1\right) \left(x + 1\right)} + \frac{3 x}{\left(x - 1\right)^{2}} - \frac{4}{x + 1} - \frac{4}{x - 1}\right)\right)}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}}$$

Simplify

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Derivative of (5x^2-2x+4)/((x-1)*(x+1))

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The solution