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The derivative of the function/ (x^2-3x)/(x-4)

Derivative of (x^2-3x)/(x-4)

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

You have entered [src] 
 2      
x  - 3*x
--------
 x - 4
x2−3xx−4\frac{x^{2} - 3 x}{x - 4} 
(x^2 - 3*x)/(x - 4)
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)=x2−3xf{\left(x \right)} = x^{2} - 3 x and g(x)=x−4g{\left(x \right)} = x - 4.

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

    1. Differentiate x2−3xx^{2} - 3 x term by term:

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

      2. 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: 2x−32 x - 3

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

    1. Differentiate x−4x - 4 term by term:

      1. The derivative of the constant −4-4 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

    −x2+3x+(x−4)(2x−3)(x−4)2\frac{- x^{2} + 3 x + \left(x - 4\right) \left(2 x - 3\right)}{\left(x - 4\right)^{2}}

  2. Now simplify:

    x2−8x+12x2−8x+16\frac{x^{2} - 8 x + 12}{x^{2} - 8 x + 16}

The answer is:

x2−8x+12x2−8x+16\frac{x^{2} - 8 x + 12}{x^{2} - 8 x + 16}

The graph
02468-8-6-4-2-1010-20001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
            2      
-3 + 2*x   x  - 3*x
-------- - --------
 x - 4            2
           (x - 4)
2x−3x−4−x2−3x(x−4)2\frac{2 x - 3}{x - 4} - \frac{x^{2} - 3 x}{\left(x - 4\right)^{2}}
Simplify
The second derivative [src] 
  /    -3 + 2*x   x*(-3 + x)\
2*|1 - -------- + ----------|
  |     -4 + x            2 |
  \               (-4 + x)  /
-----------------------------
            -4 + x
2(x(x−3)(x−4)2+1−2x−3x−4)x−4\frac{2 \left(\frac{x \left(x - 3\right)}{\left(x - 4\right)^{2}} + 1 - \frac{2 x - 3}{x - 4}\right)}{x - 4}
Simplify
The third derivative [src] 
  /     -3 + 2*x   x*(-3 + x)\
6*|-1 + -------- - ----------|
  |      -4 + x            2 |
  \                (-4 + x)  /
------------------------------
                  2           
          (-4 + x)
6(−x(x−3)(x−4)2−1+2x−3x−4)(x−4)2\frac{6 \left(- \frac{x \left(x - 3\right)}{\left(x - 4\right)^{2}} - 1 + \frac{2 x - 3}{x - 4}\right)}{\left(x - 4\right)^{2}}
Simplify
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