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x^2/(x+1)^2

Derivative of x^2/(x+1)^2

Function f() - derivative -N order at the point
v

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Piecewise:

The solution

You have entered [src]
    2   
   x    
--------
       2
(x + 1) 
x2(x+1)2\frac{x^{2}}{\left(x + 1\right)^{2}}
x^2/(x + 1)^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+1)2g{\left(x \right)} = \left(x + 1\right)^{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. Let u=x+1u = x + 1.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddx(x+1)\frac{d}{d x} \left(x + 1\right):

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

        1. The derivative of the constant 11 is zero.

        2. Apply the power rule: xx goes to 11

        The result is: 11

      The result of the chain rule is:

      2x+22 x + 2

    Now plug in to the quotient rule:

    x2(2x+2)+2x(x+1)2(x+1)4\frac{- x^{2} \left(2 x + 2\right) + 2 x \left(x + 1\right)^{2}}{\left(x + 1\right)^{4}}

  2. Now simplify:

    2xx3+3x2+3x+1\frac{2 x}{x^{3} + 3 x^{2} + 3 x + 1}


The answer is:

2xx3+3x2+3x+1\frac{2 x}{x^{3} + 3 x^{2} + 3 x + 1}

The graph
02468-8-6-4-2-1010-50005000
The first derivative [src]
            2           
  2*x      x *(-2 - 2*x)
-------- + -------------
       2             4  
(x + 1)       (x + 1)   
x2(2x2)(x+1)4+2x(x+1)2\frac{x^{2} \left(- 2 x - 2\right)}{\left(x + 1\right)^{4}} + \frac{2 x}{\left(x + 1\right)^{2}}
The second derivative [src]
  /                 2  \
  |     4*x      3*x   |
2*|1 - ----- + --------|
  |    1 + x          2|
  \            (1 + x) /
------------------------
               2        
        (1 + x)         
2(3x2(x+1)24xx+1+1)(x+1)2\frac{2 \left(\frac{3 x^{2}}{\left(x + 1\right)^{2}} - \frac{4 x}{x + 1} + 1\right)}{\left(x + 1\right)^{2}}
The third derivative [src]
   /          2          \
   |       2*x       3*x |
12*|-1 - -------- + -----|
   |            2   1 + x|
   \     (1 + x)         /
--------------------------
                3         
         (1 + x)          
12(2x2(x+1)2+3xx+11)(x+1)3\frac{12 \left(- \frac{2 x^{2}}{\left(x + 1\right)^{2}} + \frac{3 x}{x + 1} - 1\right)}{\left(x + 1\right)^{3}}
The graph
Derivative of x^2/(x+1)^2