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Identical expressions
y=(x²- two)/(x²+ two)
y equally (x² minus 2) divide by (x² plus 2)
y equally (x² minus two) divide by (x² plus two)
y=x²-2/x²+2
y=(x²-2) divide by (x²+2)
Similar expressions
y=(x²-2)/(x²-2)
y=(x²+2)/(x²+2)

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

Derivative of y=(x²-2)/(x²+2)

The solution

You have entered [src]

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

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

(x^2 - 2)/(x^2 + 2)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             / 2    \
 2*x     2*x*\x  - 2/
------ - ------------
 2                2  
x  + 2    / 2    \   
          \x  + 2/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

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Derivative of y=(x²-2)/(x²+2)

The graph:

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The solution