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Identical expressions
(x^ two - three *x+ three)/(x- two)^ two
(x squared minus 3 multiply by x plus 3) divide by (x minus 2) squared
(x to the power of two minus three multiply by x plus three) divide by (x minus two) to the power of two
(x2-3*x+3)/(x-2)2
x2-3*x+3/x-22
(x²-3*x+3)/(x-2)²
(x to the power of 2-3*x+3)/(x-2) to the power of 2
(x^2-3x+3)/(x-2)^2
(x2-3x+3)/(x-2)2
x2-3x+3/x-22
x^2-3x+3/x-2^2
(x^2-3*x+3) divide by (x-2)^2
Similar expressions
(x^2+3*x+3)/(x-2)^2
(x^2-3*x+3)/(x+2)^2
(x^2-3*x-3)/(x-2)^2

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} - 3 x + 3$ term by term:
  1. The derivative of the constant $3$ 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. 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:

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

$- \frac{x}{x^{3} - 6 x^{2} + 12 x - 8}$

The answer is:

$- \frac{x}{x^{3} - 6 x^{2} + 12 x - 8}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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Piecewise:

The solution