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Similar expressions
(10x^4-3x^2)/(3x^2+1)^2
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The derivative of the function/ (10x^4-3x^2)/(3x^2-1)^2

Derivative of (10x^4-3x^2)/(3x^2-1)^2

The solution

You have entered [src]

    4      2
10*x  - 3*x 
------------
          2 
/   2    \  
\3*x  - 1/

$$\frac{10 x^{4} - 3 x^{2}}{\left(3 x^{2} - 1\right)^{2}}$$

  /    4      2\
d |10*x  - 3*x |
--|------------|
dx|          2 |
  |/   2    \  |
  \\3*x  - 1/  /

$$\frac{d}{d x} \frac{10 x^{4} - 3 x^{2}}{\left(3 x^{2} - 1\right)^{2}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 10 x^{4} - 3 x^{2}$ and $g{\left(x \right)} = \left(3 x^{2} - 1\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $10 x^{4} - 3 x^{2}$ term by term:
  1. 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: $- 6 x$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
    So, the result is: $40 x^{3}$
  The result is: $40 x^{3} - 6 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x^{2} - 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{2} - 1\right)$ :
  1. Differentiate $3 x^{2} - 1$ term by term:
    1. The derivative of the constant $-1$ 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: $6 x$
    The result is: $6 x$
  The result of the chain rule is:
  
  $6 x \left(6 x^{2} - 2\right)$
Now plug in to the quotient rule:

$\frac{- 6 x \left(6 x^{2} - 2\right) \left(10 x^{4} - 3 x^{2}\right) + \left(3 x^{2} - 1\right)^{2} \cdot \left(40 x^{3} - 6 x\right)}{\left(3 x^{2} - 1\right)^{4}}$
Now simplify:

$\frac{- 22 x^{3} + 6 x}{27 x^{6} - 27 x^{4} + 9 x^{2} - 1}$

The answer is:

$\frac{- 22 x^{3} + 6 x}{27 x^{6} - 27 x^{4} + 9 x^{2} - 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           3        /    4      2\
-6*x + 40*x    12*x*\10*x  - 3*x /
------------ - -------------------
          2                  3    
/   2    \         /   2    \     
\3*x  - 1/         \3*x  - 1/

$$- \frac{12 x \left(10 x^{4} - 3 x^{2}\right)}{\left(3 x^{2} - 1\right)^{3}} + \frac{40 x^{3} - 6 x}{\left(3 x^{2} - 1\right)^{2}}$$

Simplify

The second derivative [src]

  /                                      /           2  \             \
  |                                    2 |       18*x   | /         2\|
  |                                 2*x *|-1 + ---------|*\-3 + 10*x /|
  |                2 /         2\        |             2|             |
  |         2   8*x *\-3 + 20*x /        \     -1 + 3*x /             |
6*|-1 + 20*x  - ----------------- + ----------------------------------|
  |                         2                           2             |
  \                 -1 + 3*x                    -1 + 3*x              /
-----------------------------------------------------------------------
                                         2                             
                              /        2\                              
                              \-1 + 3*x /

$$\frac{6 \cdot \left(\frac{2 x^{2} \cdot \left(10 x^{2} - 3\right) \left(\frac{18 x^{2}}{3 x^{2} - 1} - 1\right)}{3 x^{2} - 1} + 20 x^{2} - \frac{8 x^{2} \cdot \left(20 x^{2} - 3\right)}{3 x^{2} - 1} - 1\right)}{\left(3 x^{2} - 1\right)^{2}}$$

Simplify

The third derivative [src]

     /                        /           2  \                      /           2  \             \
     |                        |       18*x   | /         2\       2 |        8*x   | /         2\|
     |                      3*|-1 + ---------|*\-3 + 20*x /   27*x *|-1 + ---------|*\-3 + 10*x /|
     |       /         2\     |             2|                      |             2|             |
     |     9*\-1 + 20*x /     \     -1 + 3*x /                      \     -1 + 3*x /             |
24*x*|10 - -------------- + ------------------------------- - -----------------------------------|
     |               2                         2                                     2           |
     |       -1 + 3*x                  -1 + 3*x                           /        2\            |
     \                                                                    \-1 + 3*x /            /
--------------------------------------------------------------------------------------------------
                                                      2                                           
                                           /        2\                                            
                                           \-1 + 3*x /

$$\frac{24 x \left(- \frac{27 x^{2} \cdot \left(10 x^{2} - 3\right) \left(\frac{8 x^{2}}{3 x^{2} - 1} - 1\right)}{\left(3 x^{2} - 1\right)^{2}} + \frac{3 \cdot \left(20 x^{2} - 3\right) \left(\frac{18 x^{2}}{3 x^{2} - 1} - 1\right)}{3 x^{2} - 1} + 10 - \frac{9 \cdot \left(20 x^{2} - 1\right)}{3 x^{2} - 1}\right)}{\left(3 x^{2} - 1\right)^{2}}$$

Simplify

The graph

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Derivative of (10x^4-3x^2)/(3x^2-1)^2

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Piecewise:

The solution