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Derivative of:
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Identical expressions
y=(x²- four)²/(2x²+ one)³
y equally (x² minus 4)² divide by (2x² plus 1)³
y equally (x² minus four)² divide by (2x² plus one)³
y=x²-4²/2x²+1³
y=(x²-4)² divide by (2x²+1)³
Similar expressions
y=(x²-4)²/(2x²-1)³
y=(x²+4)²/(2x²+1)³

The derivative of the function/ y=(x²-4)²/(2x²+1)³

Derivative of y=(x²-4)²/(2x²+1)³

The solution

You have entered [src]

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

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

  /         2 \
  | / 2    \  |
d | \x  - 4/  |
--|-----------|
dx|          3|
  |/   2    \ |
  \\2*x  + 1/ /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(x^{2} - 4\right)^{2}$ and $g{\left(x \right)} = \left(2 x^{2} + 1\right)^{3}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{2} - 4$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 4\right)$ :
  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: $2 x$
  The result of the chain rule is:
  
  $2 x \left(2 x^{2} - 8\right)$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x^{2} + 1$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{2} + 1\right)$ :
  1. Differentiate $2 x^{2} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    The result is: $4 x$
  The result of the chain rule is:
  
  $12 x \left(2 x^{2} + 1\right)^{2}$
Now plug in to the quotient rule:

$\frac{- 12 x \left(x^{2} - 4\right)^{2} \left(2 x^{2} + 1\right)^{2} + 2 x \left(2 x^{2} - 8\right) \left(2 x^{2} + 1\right)^{3}}{\left(2 x^{2} + 1\right)^{6}}$
Now simplify:

$\frac{4 x \left(13 - x^{2}\right) \left(x^{2} - 4\right)}{\left(2 x^{2} + 1\right)^{4}}$

The answer is:

$\frac{4 x \left(13 - x^{2}\right) \left(x^{2} - 4\right)}{\left(2 x^{2} + 1\right)^{4}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$- \frac{12 x \left(x^{2} - 4\right)^{2}}{\left(2 x^{2} + 1\right)^{4}} + \frac{4 x \left(x^{2} - 4\right)}{\left(2 x^{2} + 1\right)^{3}}$$

Simplify

The second derivative [src]

  /                                         2 /          2  \\
  |                                /      2\  |      16*x   ||
  |                              3*\-4 + x / *|-1 + --------||
  |                2 /      2\                |            2||
  |        2   24*x *\-4 + x /                \     1 + 2*x /|
4*|-4 + 3*x  - --------------- + ----------------------------|
  |                       2                       2          |
  \                1 + 2*x                 1 + 2*x           /
--------------------------------------------------------------
                                   3                          
                         /       2\                           
                         \1 + 2*x /

$$\frac{4 \left(- \frac{24 x^{2} \left(x^{2} - 4\right)}{2 x^{2} + 1} + 3 x^{2} + \frac{3 \left(x^{2} - 4\right)^{2} \cdot \left(\frac{16 x^{2}}{2 x^{2} + 1} - 1\right)}{2 x^{2} + 1} - 4\right)}{\left(2 x^{2} + 1\right)^{3}}$$

Simplify

The third derivative [src]

     /                               2 /          2  \     /          2  \          \
     |                      /      2\  |      20*x   |     |      16*x   | /      2\|
     |                    8*\-4 + x / *|-3 + --------|   6*|-1 + --------|*\-4 + x /|
     |      /        2\                |            2|     |            2|          |
     |    6*\-4 + 3*x /                \     1 + 2*x /     \     1 + 2*x /          |
24*x*|1 - ------------- - ---------------------------- + ---------------------------|
     |              2                       2                             2         |
     |       1 + 2*x              /       2\                       1 + 2*x          |
     \                            \1 + 2*x /                                        /
-------------------------------------------------------------------------------------
                                               3                                     
                                     /       2\                                      
                                     \1 + 2*x /

$$\frac{24 x \left(- \frac{8 \left(x^{2} - 4\right)^{2} \cdot \left(\frac{20 x^{2}}{2 x^{2} + 1} - 3\right)}{\left(2 x^{2} + 1\right)^{2}} + \frac{6 \left(x^{2} - 4\right) \left(\frac{16 x^{2}}{2 x^{2} + 1} - 1\right)}{2 x^{2} + 1} + 1 - \frac{6 \cdot \left(3 x^{2} - 4\right)}{2 x^{2} + 1}\right)}{\left(2 x^{2} + 1\right)^{3}}$$

Simplify

The graph

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Derivative of y=(x²-4)²/(2x²+1)³

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Piecewise:

The solution