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The derivative of the function/ (x^2-6x+13)/(x-3)

Derivative of (x^2-6x+13)/(x-3)

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

You have entered [src] 
 2           
x  - 6*x + 13
-------------
    x - 3
(x2−6x)+13x−3\frac{\left(x^{2} - 6 x\right) + 13}{x - 3} 
(x^2 - 6*x + 13)/(x - 3)
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)=x2−6x+13f{\left(x \right)} = x^{2} - 6 x + 13 and g(x)=x−3g{\left(x \right)} = x - 3.

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

    1. Differentiate x2−6x+13x^{2} - 6 x + 13 term by term:

      1. The derivative of the constant 1313 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: −6-6

      The result is: 2x−62 x - 6

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

    1. Differentiate x−3x - 3 term by term:

      1. The derivative of the constant −3-3 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

    −x2+6x+(x−3)(2x−6)−13(x−3)2\frac{- x^{2} + 6 x + \left(x - 3\right) \left(2 x - 6\right) - 13}{\left(x - 3\right)^{2}}

  2. Now simplify:

    x2−6x+5x2−6x+9\frac{x^{2} - 6 x + 5}{x^{2} - 6 x + 9}

The answer is:

x2−6x+5x2−6x+9\frac{x^{2} - 6 x + 5}{x^{2} - 6 x + 9}

The graph
02468-8-6-4-2-1010-10001000
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Plot the graph f'(x)
The first derivative [src] 
            2           
-6 + 2*x   x  - 6*x + 13
-------- - -------------
 x - 3               2  
              (x - 3)
2x−6x−3−(x2−6x)+13(x−3)2\frac{2 x - 6}{x - 3} - \frac{\left(x^{2} - 6 x\right) + 13}{\left(x - 3\right)^{2}}
Simplify
The second derivative [src] 
  /           2      \
  |     13 + x  - 6*x|
2*|-1 + -------------|
  |               2  |
  \       (-3 + x)   /
----------------------
        -3 + x
2(−1+x2−6x+13(x−3)2)x−3\frac{2 \left(-1 + \frac{x^{2} - 6 x + 13}{\left(x - 3\right)^{2}}\right)}{x - 3}
Simplify
The third derivative [src] 
  /          2      \
  |    13 + x  - 6*x|
6*|1 - -------------|
  |              2  |
  \      (-3 + x)   /
---------------------
              2      
      (-3 + x)
6(1−x2−6x+13(x−3)2)(x−3)2\frac{6 \left(1 - \frac{x^{2} - 6 x + 13}{\left(x - 3\right)^{2}}\right)}{\left(x - 3\right)^{2}}
Simplify
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