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Derivative of (x²-3x+3)/(x-1)

Function f() - derivative -N order at the point
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The solution

You have entered [src]
 2          
x  - 3*x + 3
------------
   x - 1    
(x23x)+3x1\frac{\left(x^{2} - 3 x\right) + 3}{x - 1}
(x^2 - 3*x + 3)/(x - 1)
Detail solution
  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x23x+3f{\left(x \right)} = x^{2} - 3 x + 3 and g(x)=x1g{\left(x \right)} = x - 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate x23x+3x^{2} - 3 x + 3 term by term:

      1. The derivative of the constant 33 is zero.

      2. Apply the power rule: x2x^{2} goes to 2x2 x

      3. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 3-3

      The result is: 2x32 x - 3

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x1x - 1 term by term:

      1. The derivative of the constant 1-1 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

    x2+3x+(x1)(2x3)3(x1)2\frac{- x^{2} + 3 x + \left(x - 1\right) \left(2 x - 3\right) - 3}{\left(x - 1\right)^{2}}

  2. Now simplify:

    x(x2)x22x+1\frac{x \left(x - 2\right)}{x^{2} - 2 x + 1}


The answer is:

x(x2)x22x+1\frac{x \left(x - 2\right)}{x^{2} - 2 x + 1}

The graph
02468-8-6-4-2-1010-200100
The first derivative [src]
            2          
-3 + 2*x   x  - 3*x + 3
-------- - ------------
 x - 1              2  
             (x - 1)   
2x3x1(x23x)+3(x1)2\frac{2 x - 3}{x - 1} - \frac{\left(x^{2} - 3 x\right) + 3}{\left(x - 1\right)^{2}}
The second derivative [src]
  /         2                 \
  |    3 + x  - 3*x   -3 + 2*x|
2*|1 + ------------ - --------|
  |             2      -1 + x |
  \     (-1 + x)              /
-------------------------------
             -1 + x            
2(12x3x1+x23x+3(x1)2)x1\frac{2 \left(1 - \frac{2 x - 3}{x - 1} + \frac{x^{2} - 3 x + 3}{\left(x - 1\right)^{2}}\right)}{x - 1}
The third derivative [src]
  /                     2      \
  |     -3 + 2*x   3 + x  - 3*x|
6*|-1 + -------- - ------------|
  |      -1 + x             2  |
  \                 (-1 + x)   /
--------------------------------
                   2            
           (-1 + x)             
6(1+2x3x1x23x+3(x1)2)(x1)2\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}}