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Identical expressions
-1 two x/(nine -x^ two)^2
minus 12x divide by (9 minus x squared ) squared
minus 1 two x divide by (nine minus x to the power of two) squared
-12x/(9-x2)2
-12x/9-x22
-12x/(9-x²)²
-12x/(9-x to the power of 2) to the power of 2
-12x/9-x^2^2
-12x divide by (9-x^2)^2
Similar expressions
12x/(9-x^2)^2
-12x/(9+x^2)^2

The derivative of the function/ -12x/(9-x^2)^2

Derivative of -12x/(9-x^2)^2

The solution

You have entered [src]

  -12*x  
---------
        2
/     2\ 
\9 - x /

$$\frac{\left(-1\right) 12 x}{\left(9 - x^{2}\right)^{2}}$$

(-12*x)/(9 - x^2)^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - 12 x$ and $g{\left(x \right)} = \left(9 - x^{2}\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $-12$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 9 - x^{2}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(9 - x^{2}\right)$ :
  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: $- 2 x$
  The result of the chain rule is:
  
  $- 2 x \left(18 - 2 x^{2}\right)$
Now plug in to the quotient rule:

$\frac{- 24 x^{2} \left(18 - 2 x^{2}\right) - 12 \left(9 - x^{2}\right)^{2}}{\left(9 - x^{2}\right)^{4}}$
Now simplify:

$\frac{36 \left(x^{2} + 3\right)}{\left(x^{2} - 9\right)^{3}}$

The answer is:

$\frac{36 \left(x^{2} + 3\right)}{\left(x^{2} - 9\right)^{3}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

                    2  
      12        48*x   
- --------- - ---------
          2           3
  /     2\    /     2\ 
  \9 - x /    \9 - x /

$$- \frac{48 x^{2}}{\left(9 - x^{2}\right)^{3}} - \frac{12}{\left(9 - x^{2}\right)^{2}}$$

Simplify

The second derivative [src]

     /         2 \
     |      6*x  |
48*x*|3 - -------|
     |          2|
     \    -9 + x /
------------------
             3    
    /      2\     
    \-9 + x /

$$\frac{48 x \left(- \frac{6 x^{2}}{x^{2} - 9} + 3\right)}{\left(x^{2} - 9\right)^{3}}$$

Simplify

The third derivative [src]

    /                   /          2 \\
    |                 2 |       8*x  ||
    |              2*x *|-3 + -------||
    |         2         |           2||
    |      6*x          \     -9 + x /|
144*|1 - ------- + -------------------|
    |          2               2      |
    \    -9 + x          -9 + x       /
---------------------------------------
                        3              
               /      2\               
               \-9 + x /

$$\frac{144 \left(\frac{2 x^{2} \left(\frac{8 x^{2}}{x^{2} - 9} - 3\right)}{x^{2} - 9} - \frac{6 x^{2}}{x^{2} - 9} + 1\right)}{\left(x^{2} - 9\right)^{3}}$$

Simplify

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Derivative of -12x/(9-x^2)^2

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The solution