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Integral of d{x}:
x^2/(x^2+4)
Identical expressions
x^ two /(x^ two + four)
x squared divide by (x squared plus 4)
x to the power of two divide by (x to the power of two plus four)
x2/(x2+4)
x2/x2+4
x²/(x²+4)
x to the power of 2/(x to the power of 2+4)
x^2/x^2+4
x^2 divide by (x^2+4)
Similar expressions
x^2/(x^2-4)

The derivative of the function/ x^2/(x^2+4)

Derivative of x^2/(x^2+4)

The solution

You have entered [src]

   2  
  x   
------
 2    
x  + 4

$$\frac{x^{2}}{x^{2} + 4}$$

x^2/(x^2 + 4)

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 $g{\left(x \right)} = x^{2} + 4$ .

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 $x^{2} + 4$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

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

$\frac{8 x}{\left(x^{2} + 4\right)^{2}}$

The answer is:

$\frac{8 x}{\left(x^{2} + 4\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     /                   /         2 \\
     |                 2 |      2*x  ||
     |              2*x *|-1 + ------||
     |         2         |          2||
     |      4*x          \     4 + x /|
12*x*|-2 + ------ - ------------------|
     |          2              2      |
     \     4 + x          4 + x       /
---------------------------------------
                       2               
               /     2\                
               \4 + x /

$$\frac{12 x \left(- \frac{2 x^{2} \left(\frac{2 x^{2}}{x^{2} + 4} - 1\right)}{x^{2} + 4} + \frac{4 x^{2}}{x^{2} + 4} - 2\right)}{\left(x^{2} + 4\right)^{2}}$$

Simplify

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Derivative of x^2/(x^2+4)

The graph:

Piecewise:

The solution