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Derivative of:
Derivative of (-2)/x^2
Derivative of 2x^2-x
Derivative of (cos(x))^sin(x)
Derivative of y=3x^2
Graphing y =:
-2x/(x^2-1)^2
Identical expressions
- two x/(x^ two - one)^2
minus 2x divide by (x squared minus 1) squared
minus two x divide by (x to the power of two minus one) squared
-2x/(x2-1)2
-2x/x2-12
-2x/(x²-1)²
-2x/(x to the power of 2-1) to the power of 2
-2x/x^2-1^2
-2x divide by (x^2-1)^2
Similar expressions
-2x/(x^2+1)^2
2x/(x^2-1)^2

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

Derivative of -2x/(x^2-1)^2

The solution

You have entered [src]

   -2*x  
---------
        2
/ 2    \ 
\x  - 1/

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

d /   -2*x  \
--|---------|
dx|        2|
  |/ 2    \ |
  \\x  - 1/ /

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

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x$ and $g{\left(x \right)} = \left(x^{2} - 1\right)^{2}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  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:
  
  $\frac{- 2 x^{2} \cdot \left(2 x^{2} - 2\right) + \left(x^{2} - 1\right)^{2}}{\left(x^{2} - 1\right)^{4}}$
So, the result is: $- \frac{2 \left(- 2 x^{2} \cdot \left(2 x^{2} - 2\right) + \left(x^{2} - 1\right)^{2}\right)}{\left(x^{2} - 1\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                    2  
      2          8*x   
- --------- + ---------
          2           3
  / 2    \    / 2    \ 
  \x  - 1/    \x  - 1/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of -2x/(x^2-1)^2

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Piecewise:

The solution