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Integral of d{x}:
x^2/(1+x^3)
Limit of the function:
x^2/(1+x^3)
Identical expressions
x^ two /(one +x^ three)
x squared divide by (1 plus x cubed )
x to the power of two divide by (one plus x to the power of three)
x2/(1+x3)
x2/1+x3
x²/(1+x³)
x to the power of 2/(1+x to the power of 3)
x^2/1+x^3
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Similar expressions
x^2/(1-x^3)

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

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

The solution

You have entered [src]

   2  
  x   
------
     3
1 + x

$$\frac{x^{2}}{x^{3} + 1}$$

x^2/(1 + x^3)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        4           
     3*x       2*x  
- --------- + ------
          2        3
  /     3\    1 + x 
  \1 + x /

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

Simplify

The second derivative [src]

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

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

Simplify

3-я производная [src]

     /            6         3 \
   2 |        27*x      36*x  |
6*x *|-10 - --------- + ------|
     |              2        3|
     |      /     3\    1 + x |
     \      \1 + x /          /
-------------------------------
                   2           
           /     3\            
           \1 + x /

$$\frac{6 x^{2} \left(- \frac{27 x^{6}}{\left(x^{3} + 1\right)^{2}} + \frac{36 x^{3}}{x^{3} + 1} - 10\right)}{\left(x^{3} + 1\right)^{2}}$$

Simplify

The third derivative [src]

     /            6         3 \
   2 |        27*x      36*x  |
6*x *|-10 - --------- + ------|
     |              2        3|
     |      /     3\    1 + x |
     \      \1 + x /          /
-------------------------------
                   2           
           /     3\            
           \1 + x /

$$\frac{6 x^{2} \left(- \frac{27 x^{6}}{\left(x^{3} + 1\right)^{2}} + \frac{36 x^{3}}{x^{3} + 1} - 10\right)}{\left(x^{3} + 1\right)^{2}}$$

Simplify

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

The graph:

Piecewise:

The solution