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Identical expressions
(x²- three x+3)/(x- one)
(x² minus 3x plus 3) divide by (x minus 1)
(x² minus three x plus 3) divide by (x minus one)
x²-3x+3/x-1
(x²-3x+3) divide by (x-1)
Similar expressions
(x²-3x-3)/(x-1)
(x²+3x+3)/(x-1)
(x²-3x+3)/(x+1)

The derivative of the function/ (x²-3x+3)/(x-1)

Derivative of (x²-3x+3)/(x-1)

The solution

You have entered [src]

 2          
x  - 3*x + 3
------------
   x - 1

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

(x^2 - 3*x + 3)/(x - 1)

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 $g{\left(x \right)} = x - 1$ .

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. Differentiate $x - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            2          
-3 + 2*x   x  - 3*x + 3
-------- - ------------
 x - 1              2  
             (x - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (x²-3x+3)/(x-1)

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The solution