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The derivative of the function/ (x^2+5)/(x-3)

Derivative of (x^2+5)/(x-3)

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

You have entered [src] 
 2    
x  + 5
------
x - 3
x2+5x−3\frac{x^{2} + 5}{x - 3} 
(x^2 + 5)/(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+5f{\left(x \right)} = x^{2} + 5 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+5x^{2} + 5 term by term:

      1. The derivative of the constant 55 is zero.

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

      The result is: 2x2 x

    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+2x(x−3)−5(x−3)2\frac{- x^{2} + 2 x \left(x - 3\right) - 5}{\left(x - 3\right)^{2}}

The answer is:

−x2+2x(x−3)−5(x−3)2\frac{- x^{2} + 2 x \left(x - 3\right) - 5}{\left(x - 3\right)^{2}}

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    2             
   x  + 5     2*x 
- -------- + -----
         2   x - 3
  (x - 3)
2xx−3−x2+5(x−3)2\frac{2 x}{x - 3} - \frac{x^{2} + 5}{\left(x - 3\right)^{2}}
Simplify
The second derivative [src] 
  /           2          \
  |      5 + x      2*x  |
2*|1 + --------- - ------|
  |            2   -3 + x|
  \    (-3 + x)          /
--------------------------
          -3 + x
2(−2xx−3+1+x2+5(x−3)2)x−3\frac{2 \left(- \frac{2 x}{x - 3} + 1 + \frac{x^{2} + 5}{\left(x - 3\right)^{2}}\right)}{x - 3}
Simplify
3-я производная [src] 
  /            2          \
  |       5 + x      2*x  |
6*|-1 - --------- + ------|
  |             2   -3 + x|
  \     (-3 + x)          /
---------------------------
                 2         
         (-3 + x)
6(2xx−3−1−x2+5(x−3)2)(x−3)2\frac{6 \left(\frac{2 x}{x - 3} - 1 - \frac{x^{2} + 5}{\left(x - 3\right)^{2}}\right)}{\left(x - 3\right)^{2}}
Simplify
The third derivative [src] 
  /            2          \
  |       5 + x      2*x  |
6*|-1 - --------- + ------|
  |             2   -3 + x|
  \     (-3 + x)          /
---------------------------
                 2         
         (-3 + x)
6(2xx−3−1−x2+5(x−3)2)(x−3)2\frac{6 \left(\frac{2 x}{x - 3} - 1 - \frac{x^{2} + 5}{\left(x - 3\right)^{2}}\right)}{\left(x - 3\right)^{2}}
Simplify
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