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x^3/(x+1)^2
The derivative of the function/ x^3/(x+1)^2

Derivative of x^3/(x+1)^2

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

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The solution

You have entered [src] 
    3   
   x    
--------
       2
(x + 1)
x3(x+1)2\frac{x^{3}}{\left(x + 1\right)^{2}} 
x^3/(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)=x3f{\left(x \right)} = x^{3} 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: x3x^{3} goes to 3x23 x^{2}

    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:

    −x3(2x+2)+3x2(x+1)2(x+1)4\frac{- x^{3} \left(2 x + 2\right) + 3 x^{2} \left(x + 1\right)^{2}}{\left(x + 1\right)^{4}}

  2. Now simplify:

    x2(x+3)x3+3x2+3x+1\frac{x^{2} \left(x + 3\right)}{x^{3} + 3 x^{2} + 3 x + 1}

The answer is:

x2(x+3)x3+3x2+3x+1\frac{x^{2} \left(x + 3\right)}{x^{3} + 3 x^{2} + 3 x + 1}

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     2      3           
  3*x      x *(-2 - 2*x)
-------- + -------------
       2             4  
(x + 1)       (x + 1)
x3(−2x−2)(x+1)4+3x2(x+1)2\frac{x^{3} \left(- 2 x - 2\right)}{\left(x + 1\right)^{4}} + \frac{3 x^{2}}{\left(x + 1\right)^{2}}
Simplify
The second derivative [src] 
    /        2           \
    |       x        2*x |
6*x*|1 + -------- - -----|
    |           2   1 + x|
    \    (1 + x)         /
--------------------------
                2         
         (1 + x)
6x(x2(x+1)2−2xx+1+1)(x+1)2\frac{6 x \left(\frac{x^{2}}{\left(x + 1\right)^{2}} - \frac{2 x}{x + 1} + 1\right)}{\left(x + 1\right)^{2}}
Simplify
The third derivative [src] 
  /                 3          2  \
  |     6*x      4*x        9*x   |
6*|1 - ----- - -------- + --------|
  |    1 + x          3          2|
  \            (1 + x)    (1 + x) /
-----------------------------------
                     2             
              (1 + x)
6(−4x3(x+1)3+9x2(x+1)2−6xx+1+1)(x+1)2\frac{6 \left(- \frac{4 x^{3}}{\left(x + 1\right)^{3}} + \frac{9 x^{2}}{\left(x + 1\right)^{2}} - \frac{6 x}{x + 1} + 1\right)}{\left(x + 1\right)^{2}}
Simplify
The graph
Derivative of x^3/(x+1)^2
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