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Identical expressions
y=(x^ two -4x+ eight)/(x- two)^ two
y equally (x squared minus 4x plus 8) divide by (x minus 2) squared
y equally (x to the power of two minus 4x plus eight) divide by (x minus two) to the power of two
y=(x2-4x+8)/(x-2)2
y=x2-4x+8/x-22
y=(x²-4x+8)/(x-2)²
y=(x to the power of 2-4x+8)/(x-2) to the power of 2
y=x^2-4x+8/x-2^2
y=(x^2-4x+8) divide by (x-2)^2
Similar expressions
y=(x^2-4x-8)/(x-2)^2
y=(x^2+4x+8)/(x-2)^2
y=(x^2-4x+8)/(x+2)^2

The derivative of the function/ y=(x^2-4x+8)/(x-2)^2

Derivative of y=(x^2-4x+8)/(x-2)^2

The solution

You have entered [src]

 2          
x  - 4*x + 8
------------
         2  
  (x - 2)

$$\frac{x^{2} - 4 x + 8}{\left(x - 2\right)^{2}}$$

  / 2          \
d |x  - 4*x + 8|
--|------------|
dx|         2  |
  \  (x - 2)   /

$$\frac{d}{d x} \frac{x^{2} - 4 x + 8}{\left(x - 2\right)^{2}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} - 4 x + 8$ and $g{\left(x \right)} = \left(x - 2\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} - 4 x + 8$ term by term:
  1. The derivative of the constant $8$ 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: $-4$
  The result is: $2 x - 4$
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:

$\frac{\left(x - 2\right)^{2} \cdot \left(2 x - 4\right) - \left(2 x - 4\right) \left(x^{2} - 4 x + 8\right)}{\left(x - 2\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                     / 2          \
-4 + 2*x   (4 - 2*x)*\x  - 4*x + 8/
-------- + ------------------------
       2                  4        
(x - 2)            (x - 2)

$$\frac{2 x - 4}{\left(x - 2\right)^{2}} + \frac{\left(- 2 x + 4\right) \left(x^{2} - 4 x + 8\right)}{\left(x - 2\right)^{4}}$$

Simplify

The second derivative [src]

  /          2      \
  |     8 + x  - 4*x|
6*|-1 + ------------|
  |              2  |
  \      (-2 + x)   /
---------------------
              2      
      (-2 + x)

$$\frac{6 \left(-1 + \frac{x^{2} - 4 x + 8}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right)^{2}}$$

Simplify

The third derivative [src]

   /         2      \
   |    8 + x  - 4*x|
24*|1 - ------------|
   |             2  |
   \     (-2 + x)   /
---------------------
              3      
      (-2 + x)

$$\frac{24 \cdot \left(1 - \frac{x^{2} - 4 x + 8}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right)^{3}}$$

Simplify

The graph

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Derivative of y=(x^2-4x+8)/(x-2)^2

The graph:

Piecewise:

The solution