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

Derivative of x^2/(x+2)

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

You have entered [src] 
   2 
  x  
-----
x + 2
x2x+2\frac{x^{2}}{x + 2} 
x^2/(x + 2)
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)=x2f{\left(x \right)} = x^{2} and g(x)=x+2g{\left(x \right)} = x + 2.

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

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

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

    1. Differentiate x+2x + 2 term by term:

      1. The derivative of the constant 22 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+2)(x+2)2\frac{- x^{2} + 2 x \left(x + 2\right)}{\left(x + 2\right)^{2}}

  2. Now simplify:

    x(x+4)(x+2)2\frac{x \left(x + 4\right)}{\left(x + 2\right)^{2}}

The answer is:

x(x+4)(x+2)2\frac{x \left(x + 4\right)}{\left(x + 2\right)^{2}}

The graph
02468-8-6-4-2-1010-1000500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      2           
     x        2*x 
- -------- + -----
         2   x + 2
  (x + 2)
−x2(x+2)2+2xx+2- \frac{x^{2}}{\left(x + 2\right)^{2}} + \frac{2 x}{x + 2}
Simplify
The second derivative [src] 
  /        2           \
  |       x        2*x |
2*|1 + -------- - -----|
  |           2   2 + x|
  \    (2 + x)         /
------------------------
         2 + x
2(x2(x+2)2−2xx+2+1)x+2\frac{2 \left(\frac{x^{2}}{\left(x + 2\right)^{2}} - \frac{2 x}{x + 2} + 1\right)}{x + 2}
Simplify
The third derivative [src] 
  /         2           \
  |        x        2*x |
6*|-1 - -------- + -----|
  |            2   2 + x|
  \     (2 + x)         /
-------------------------
                2        
         (2 + x)
6(−x2(x+2)2+2xx+2−1)(x+2)2\frac{6 \left(- \frac{x^{2}}{\left(x + 2\right)^{2}} + \frac{2 x}{x + 2} - 1\right)}{\left(x + 2\right)^{2}}
Simplify
The graph
Derivative of x^2/(x+2)
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