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Graphing y =:
4x/(x^2+1)^2
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Similar expressions
4x/(x^2-1)^2

The derivative of the function/ 4x/(x^2+1)^2

Derivative of 4x/(x^2+1)^2

The solution

You have entered [src]

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

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

(4*x)/(x^2 + 1)^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 4 x$ and $g{\left(x \right)} = \left(x^{2} + 1\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $4$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{2} + 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
  1. Differentiate $x^{2} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Apply the power rule: $x^{2}$ goes to $2 x$
    The result is: $2 x$
  The result of the chain rule is:
  
  $2 x \left(2 x^{2} + 2\right)$
Now plug in to the quotient rule:

$\frac{- 8 x^{2} \left(2 x^{2} + 2\right) + 4 \left(x^{2} + 1\right)^{2}}{\left(x^{2} + 1\right)^{4}}$
Now simplify:

$\frac{4 - 12 x^{2}}{x^{6} + 3 x^{4} + 3 x^{2} + 1}$

The answer is:

$\frac{4 - 12 x^{2}}{x^{6} + 3 x^{4} + 3 x^{2} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                  2  
    4         16*x   
--------- - ---------
        2           3
/ 2    \    / 2    \ 
\x  + 1/    \x  + 1/

$$- \frac{16 x^{2}}{\left(x^{2} + 1\right)^{3}} + \frac{4}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The second derivative [src]

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

$$\frac{16 x \left(\frac{6 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{3}}$$

Simplify

The third derivative [src]

   /                   /         2 \\
   |                 2 |      8*x  ||
   |              2*x *|-3 + ------||
   |         2         |          2||
   |      6*x          \     1 + x /|
48*|-1 + ------ - ------------------|
   |          2              2      |
   \     1 + x          1 + x       /
-------------------------------------
                      3              
              /     2\               
              \1 + x /

$$\frac{48 \left(- \frac{2 x^{2} \left(\frac{8 x^{2}}{x^{2} + 1} - 3\right)}{x^{2} + 1} + \frac{6 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{3}}$$

Simplify

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Derivative of 4x/(x^2+1)^2

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The solution