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Identical expressions
-96x/(x^ two + twelve)^ two
minus 96x divide by (x squared plus 12) squared
minus 96x divide by (x to the power of two plus twelve) to the power of two
-96x/(x2+12)2
-96x/x2+122
-96x/(x²+12)²
-96x/(x to the power of 2+12) to the power of 2
-96x/x^2+12^2
-96x divide by (x^2+12)^2
Similar expressions
96x/(x^2+12)^2
-96x/(x^2-12)^2

The derivative of the function/ -96x/(x^2+12)^2

Derivative of -96x/(x^2+12)^2

The solution

You have entered [src]

  -96*x   
----------
         2
/ 2     \ 
\x  + 12/

$$- \frac{96 x}{\left(x^{2} + 12\right)^{2}}$$

d /  -96*x   \
--|----------|
dx|         2|
  |/ 2     \ |
  \\x  + 12/ /

$$\frac{d}{d x} \left(- \frac{96 x}{\left(x^{2} + 12\right)^{2}}\right)$$

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 $g{\left(x \right)} = \left(x^{2} + 12\right)^{2}$ .
  
  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 $u = x^{2} + 12$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 12\right)$ :
    1. Differentiate $x^{2} + 12$ term by term:
      1. The derivative of the constant $12$ 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:
    
    $2 x \left(2 x^{2} + 24\right)$
  Now plug in to the quotient rule:
  
  $\frac{- 2 x^{2} \cdot \left(2 x^{2} + 24\right) + \left(x^{2} + 12\right)^{2}}{\left(x^{2} + 12\right)^{4}}$
So, the result is: $- \frac{96 \left(- 2 x^{2} \cdot \left(2 x^{2} + 24\right) + \left(x^{2} + 12\right)^{2}\right)}{\left(x^{2} + 12\right)^{4}}$
Now simplify:

$\frac{288 \left(x^{2} - 4\right)}{\left(x^{2} + 12\right)^{3}}$

The answer is:

$\frac{288 \left(x^{2} - 4\right)}{\left(x^{2} + 12\right)^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      2  
      96         384*x   
- ---------- + ----------
           2            3
  / 2     \    / 2     \ 
  \x  + 12/    \x  + 12/

$$\frac{384 x^{2}}{\left(x^{2} + 12\right)^{3}} - \frac{96}{\left(x^{2} + 12\right)^{2}}$$

Simplify

The second derivative [src]

      /         2 \
      |      6*x  |
384*x*|3 - -------|
      |          2|
      \    12 + x /
-------------------
              3    
     /      2\     
     \12 + x /

$$\frac{384 x \left(- \frac{6 x^{2}}{x^{2} + 12} + 3\right)}{\left(x^{2} + 12\right)^{3}}$$

Simplify

The third derivative [src]

     /                   /          2 \\
     |                 2 |       8*x  ||
     |              2*x *|-3 + -------||
     |         2         |           2||
     |      6*x          \     12 + x /|
1152*|1 - ------- + -------------------|
     |          2               2      |
     \    12 + x          12 + x       /
----------------------------------------
                        3               
               /      2\                
               \12 + x /

$$\frac{1152 \cdot \left(\frac{2 x^{2} \cdot \left(\frac{8 x^{2}}{x^{2} + 12} - 3\right)}{x^{2} + 12} - \frac{6 x^{2}}{x^{2} + 12} + 1\right)}{\left(x^{2} + 12\right)^{3}}$$

Simplify

The graph

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Derivative of -96x/(x^2+12)^2

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The solution