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Graphing y =:
x^3/(x-2)^2
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x^ three /(x- two)^ two
x cubed divide by (x minus 2) squared
x to the power of three divide by (x minus two) to the power of two
x3/(x-2)2
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x³/(x-2)²
x to the power of 3/(x-2) to the power of 2
x^3/x-2^2
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Similar expressions
x^3/(x+2)^2

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

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

The solution

You have entered [src]

    3   
   x    
--------
       2
(x - 2)

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

x^3/(x - 2)^2

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 2$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
  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: $1$
  The result of the chain rule is:
  
  $2 x - 4$
Now plug in to the quotient rule:

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

$\frac{x^{2} \left(x - 6\right)}{x^{3} - 6 x^{2} + 12 x - 8}$

The answer is:

$\frac{x^{2} \left(x - 6\right)}{x^{3} - 6 x^{2} + 12 x - 8}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2      3          
  3*x      x *(4 - 2*x)
-------- + ------------
       2            4  
(x - 2)      (x - 2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                   3           2  \
  |     6*x        4*x         9*x   |
6*|1 - ------ - --------- + ---------|
  |    -2 + x           3           2|
  \             (-2 + x)    (-2 + x) /
--------------------------------------
                      2               
              (-2 + x)

$$\frac{6 \left(- \frac{4 x^{3}}{\left(x - 2\right)^{3}} + \frac{9 x^{2}}{\left(x - 2\right)^{2}} - \frac{6 x}{x - 2} + 1\right)}{\left(x - 2\right)^{2}}$$

Simplify

The graph

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

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The solution