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Integral of d{x}:
x/(9-x^2)
Limit of the function:
x/(9-x^2)
Identical expressions
x/(nine -x^ two)
x divide by (9 minus x squared )
x divide by (nine minus x to the power of two)
x/(9-x2)
x/9-x2
x/(9-x²)
x/(9-x to the power of 2)
x/9-x^2
x divide by (9-x^2)
Similar expressions
(-x)/(9-x^2)^(3/2)
x/(9+x^2)

The derivative of the function/ x/(9-x^2)

Derivative of x/(9-x^2)

The solution

You have entered [src]

  x   
------
     2
9 - x

$$\frac{x}{9 - x^{2}}$$

x/(9 - x^2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x$ and $g{\left(x \right)} = 9 - x^{2}$ .

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. Differentiate $9 - x^{2}$ term by term:
  1. The derivative of the constant $9$ 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: $- 2 x$
  The result is: $- 2 x$
Now plug in to the quotient rule:

$\frac{x^{2} + 9}{\left(9 - x^{2}\right)^{2}}$
Now simplify:

$\frac{x^{2} + 9}{\left(x^{2} - 9\right)^{2}}$

The answer is:

$\frac{x^{2} + 9}{\left(x^{2} - 9\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               2  
  1         2*x   
------ + ---------
     2           2
9 - x    /     2\ 
         \9 - x /

$$\frac{2 x^{2}}{\left(9 - x^{2}\right)^{2}} + \frac{1}{9 - x^{2}}$$

Simplify

The second derivative [src]

    /         2 \
    |      4*x  |
2*x*|3 - -------|
    |          2|
    \    -9 + x /
-----------------
             2   
    /      2\    
    \-9 + x /

$$\frac{2 x \left(- \frac{4 x^{2}}{x^{2} - 9} + 3\right)}{\left(x^{2} - 9\right)^{2}}$$

Simplify

The third derivative [src]

  /                   /          2 \\
  |                 2 |       2*x  ||
  |              4*x *|-1 + -------||
  |         2         |           2||
  |      4*x          \     -9 + x /|
6*|1 - ------- + -------------------|
  |          2               2      |
  \    -9 + x          -9 + x       /
-------------------------------------
                       2             
              /      2\              
              \-9 + x /

$$\frac{6 \left(\frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{4 x^{2}}{x^{2} - 9} + 1\right)}{\left(x^{2} - 9\right)^{2}}$$

Simplify

The graph

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Derivative of x/(9-x^2)

The graph:

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The solution