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Derivative of x^2/(x^2-3)
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Derivative of f(x)=1-3x^2
Derivative of cos^x(x)
Identical expressions
x^ two /(x^ two - three)
x squared divide by (x squared minus 3)
x to the power of two divide by (x to the power of two minus three)
x2/(x2-3)
x2/x2-3
x²/(x²-3)
x to the power of 2/(x to the power of 2-3)
x^2/x^2-3
x^2 divide by (x^2-3)
Similar expressions
x^2/(x^2+3)

The derivative of the function/ x^2/(x^2-3)

Derivative of x^2/(x^2-3)

The solution

You have entered [src]

   2  
  x   
------
 2    
x  - 3

$$\frac{x^{2}}{x^{2} - 3}$$

x^2/(x^2 - 3)

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 $g{\left(x \right)} = x^{2} - 3$ .

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 $x^{2} - 3$ term by term:
  1. The derivative of the constant $-3$ 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:

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

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

The answer is:

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

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  - 3
  \x  - 3/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

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Derivative of x^2/(x^2-3)

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The solution