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(- six x^ two + one 0x)6/(x^ two +1)^ four
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(-6x2+10x)6/(x2+1)4
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Similar expressions
(-6x^2+10x)6/(x^2-1)^4
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(6x^2+10x)6/(x^2+1)^4

The derivative of the function/ (-6x^2+10x)6/(x^2+1)^4

Derivative of (-6x^2+10x)6/(x^2+1)^4

The solution

You have entered [src]

/     2       \  
\- 6*x  + 10*x/*6
-----------------
            4    
    / 2    \     
    \x  + 1/

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

((-6*x^2 + 10*x)*6)/(x^2 + 1)^4

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - 36 x^{2} + 60 x$ and $g{\left(x \right)} = \left(x^{2} + 1\right)^{4}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- 36 x^{2} + 60 x$ 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: $- 72 x$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $60$
  The result is: $60 - 72 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{2} + 1$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
  1. Differentiate $x^{2} + 1$ term by term:
    1. The derivative of the constant $1$ 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:
  
  $8 x \left(x^{2} + 1\right)^{3}$
Now plug in to the quotient rule:

$\frac{- 8 x \left(- 36 x^{2} + 60 x\right) \left(x^{2} + 1\right)^{3} + \left(60 - 72 x\right) \left(x^{2} + 1\right)^{4}}{\left(x^{2} + 1\right)^{8}}$
Now simplify:

$\frac{12 \left(8 x^{2} \left(3 x - 5\right) + \left(5 - 6 x\right) \left(x^{2} + 1\right)\right)}{\left(x^{2} + 1\right)^{5}}$

The answer is:

$\frac{12 \left(8 x^{2} \left(3 x - 5\right) + \left(5 - 6 x\right) \left(x^{2} + 1\right)\right)}{\left(x^{2} + 1\right)^{5}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 /     2       \
60 - 72*x   48*x*\- 6*x  + 10*x/
--------- - --------------------
        4                5      
/ 2    \         / 2    \       
\x  + 1/         \x  + 1/

$$- \frac{48 x \left(- 6 x^{2} + 10 x\right)}{\left(x^{2} + 1\right)^{5}} + \frac{60 - 72 x}{\left(x^{2} + 1\right)^{4}}$$

Simplify

The second derivative [src]

   /                          /         2 \           \
   |                          |     10*x  |           |
   |                      4*x*|-1 + ------|*(-5 + 3*x)|
   |                          |          2|           |
   |     8*x*(-5 + 6*x)       \     1 + x /           |
24*|-3 + -------------- - ----------------------------|
   |              2                       2           |
   \         1 + x                   1 + x            /
-------------------------------------------------------
                               4                       
                       /     2\                        
                       \1 + x /

$$\frac{24 \left(- \frac{4 x \left(3 x - 5\right) \left(\frac{10 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} + \frac{8 x \left(6 x - 5\right)}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{4}}$$

Simplify

The third derivative [src]

    /                                       /         2 \           \
    |                                     2 |      4*x  |           |
    |                                 10*x *|-1 + ------|*(-5 + 3*x)|
    |      /         2 \                    |          2|           |
    |      |     10*x  |                    \     1 + x /           |
288*|6*x - |-1 + ------|*(-5 + 6*x) + ------------------------------|
    |      |          2|                               2            |
    \      \     1 + x /                          1 + x             /
---------------------------------------------------------------------
                                      5                              
                              /     2\                               
                              \1 + x /

$$\frac{288 \left(\frac{10 x^{2} \left(3 x - 5\right) \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} + 6 x - \left(6 x - 5\right) \left(\frac{10 x^{2}}{x^{2} + 1} - 1\right)\right)}{\left(x^{2} + 1\right)^{5}}$$

Simplify

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Derivative of (-6x^2+10x)6/(x^2+1)^4

The graph:

Piecewise:

The solution