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Identical expressions
(x^ three + two x)/(x-2)
(x cubed plus 2x) divide by (x minus 2)
(x to the power of three plus two x) divide by (x minus 2)
(x3+2x)/(x-2)
x3+2x/x-2
(x³+2x)/(x-2)
(x to the power of 3+2x)/(x-2)
x^3+2x/x-2
(x^3+2x) divide by (x-2)
Similar expressions
(x^3-2x)/(x-2)
(x^3+2x)/(x+2)

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

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

The solution

You have entered [src]

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

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

(x^3 + 2*x)/(x - 2)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{3} + 2 x$ term by term:
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  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: $2$
  The result is: $3 x^{2} + 2$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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The solution