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Graphing y =:
(2x^(2)-6)/(x-2)
Identical expressions
(two x^(two)- six)/(x-2)
(2x to the power of (2) minus 6) divide by (x minus 2)
(two x to the power of (two) minus six) divide by (x minus 2)
(2x(2)-6)/(x-2)
2x2-6/x-2
2x^2-6/x-2
(2x^(2)-6) divide by (x-2)
Similar expressions
(2x^(2)+6)/(x-2)
(2x^(2)-6)/(x+2)

The derivative of the function/ (2x^(2)-6)/(x-2)

Derivative of (2x^(2)-6)/(x-2)

The solution

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   2    
2*x  - 6
--------
 x - 2

$$\frac{2 x^{2} - 6}{x - 2}$$

(2*x^2 - 6)/(x - 2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x^{2} - 6$ and $g{\left(x \right)} = x - 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x^{2} - 6$ term by term:
  1. The derivative of the constant $-6$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $4 x$
  The result is: $4 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x - 2$ term by term:
  1. The derivative of the constant $-2$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{- 2 x^{2} + 4 x \left(x - 2\right) + 6}{\left(x - 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2            
  2*x  - 6    4*x 
- -------- + -----
         2   x - 2
  (x - 2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /            2          \
   |      -3 + x      2*x  |
12*|-1 - --------- + ------|
   |             2   -2 + x|
   \     (-2 + x)          /
----------------------------
                 2          
         (-2 + x)

$$\frac{12 \left(\frac{2 x}{x - 2} - 1 - \frac{x^{2} - 3}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right)^{2}}$$

Simplify

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Derivative of (2x^(2)-6)/(x-2)

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The solution