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Integral of d{x}:
x^3/(x+1)
Graphing y =:
x^3/(x+1)
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Similar expressions
x^3/(x-1)

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

Derivative of x^3/(x+1)

The solution

You have entered [src]

   3 
  x  
-----
x + 1

$$\frac{x^{3}}{x + 1}$$

x^3/(x + 1)

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 $g{\left(x \right)} = x + 1$ .

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. 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: $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

    /        2           \
    |       x        3*x |
2*x*|3 + -------- - -----|
    |           2   1 + x|
    \    (1 + x)         /
--------------------------
          1 + x

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

Simplify

The third derivative [src]

  /        3                   2  \
  |       x        3*x      3*x   |
6*|1 - -------- - ----- + --------|
  |           3   1 + x          2|
  \    (1 + x)            (1 + x) /
-----------------------------------
               1 + x

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

Simplify

The graph

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Derivative of x^3/(x+1)

The graph:

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The solution