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Derivative of:
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Graphing y =:
x^3/(x+1)^2
Identical expressions
x^ three /(x+ one)^ two
x cubed divide by (x plus 1) squared
x to the power of three divide by (x plus one) to the power of two
x3/(x+1)2
x3/x+12
x³/(x+1)²
x to the power of 3/(x+1) to the power of 2
x^3/x+1^2
x^3 divide by (x+1)^2
Similar expressions
x^3/(x-1)^2

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

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

The solution

You have entered [src]

    3   
   x    
--------
       2
(x + 1)

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

x^3/(x + 1)^2

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{3}$ and $g{\left(x \right)} = \left(x + 1\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
  1. Differentiate $x + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x + 2$
Now plug in to the quotient rule:

$\frac{- x^{3} \left(2 x + 2\right) + 3 x^{2} \left(x + 1\right)^{2}}{\left(x + 1\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

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)

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

Simplify

The second derivative [src]

    /        2           \
    |       x        2*x |
6*x*|1 + -------- - -----|
    |           2   1 + x|
    \    (1 + x)         /
--------------------------
                2         
         (1 + x)

$$\frac{6 x \left(\frac{x^{2}}{\left(x + 1\right)^{2}} - \frac{2 x}{x + 1} + 1\right)}{\left(x + 1\right)^{2}}$$

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)

$$\frac{6 \left(- \frac{4 x^{3}}{\left(x + 1\right)^{3}} + \frac{9 x^{2}}{\left(x + 1\right)^{2}} - \frac{6 x}{x + 1} + 1\right)}{\left(x + 1\right)^{2}}$$

Simplify

The graph

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

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The solution