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Identical expressions
(three x^ four - two x+ five)/(x^2+3)
(3x to the power of 4 minus 2x plus 5) divide by (x squared plus 3)
(three x to the power of four minus two x plus five) divide by (x squared plus 3)
(3x4-2x+5)/(x2+3)
3x4-2x+5/x2+3
(3x⁴-2x+5)/(x²+3)
(3x to the power of 4-2x+5)/(x to the power of 2+3)
3x^4-2x+5/x^2+3
(3x^4-2x+5) divide by (x^2+3)
Similar expressions
(3x^4-2x+5)/(x^2-3)
(3x^4+2x+5)/(x^2+3)
(3x^4-2x-5)/(x^2+3)

The derivative of the function/ (3x^4-2x+5)/(x^2+3)

Derivative of (3x^4-2x+5)/(x^2+3)

The solution

You have entered [src]

   4          
3*x  - 2*x + 5
--------------
     2        
    x  + 3

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

(3*x^4 - 2*x + 5)/(x^2 + 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 x^{4} - 2 x + 5$ and $g{\left(x \right)} = x^{2} + 3$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x^{4} - 2 x + 5$ term by term:
  1. The derivative of the constant $5$ 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: $x$ goes to $1$
    So, the result is: $-2$
  3. 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: $12 x^{3}$
  The result is: $12 x^{3} - 2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + 3$ term by term:
  1. The derivative of the constant $3$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

$\frac{- 2 x \left(3 x^{4} - 2 x + 5\right) + \left(x^{2} + 3\right) \left(12 x^{3} - 2\right)}{\left(x^{2} + 3\right)^{2}}$
Now simplify:

$\frac{2 \left(3 x^{5} + 18 x^{3} + x^{2} - 5 x - 3\right)}{x^{4} + 6 x^{2} + 9}$

The answer is:

$\frac{2 \left(3 x^{5} + 18 x^{3} + x^{2} - 5 x - 3\right)}{x^{4} + 6 x^{2} + 9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         3       /   4          \
-2 + 12*x    2*x*\3*x  - 2*x + 5/
---------- - --------------------
   2                      2      
  x  + 3          / 2    \       
                  \x  + 3/

$$- \frac{2 x \left(\left(3 x^{4} - 2 x\right) + 5\right)}{\left(x^{2} + 3\right)^{2}} + \frac{12 x^{3} - 2}{x^{2} + 3}$$

Simplify

The second derivative [src]

  /        /         2 \                                   \
  |        |      4*x  | /             4\                  |
  |        |-1 + ------|*\5 - 2*x + 3*x /                  |
  |        |          2|                        /        3\|
  |    2   \     3 + x /                    4*x*\-1 + 6*x /|
2*|18*x  + ------------------------------ - ---------------|
  |                         2                         2    |
  \                    3 + x                     3 + x     /
------------------------------------------------------------
                                2                           
                           3 + x

$$\frac{2 \left(18 x^{2} - \frac{4 x \left(6 x^{3} - 1\right)}{x^{2} + 3} + \frac{\left(\frac{4 x^{2}}{x^{2} + 3} - 1\right) \left(3 x^{4} - 2 x + 5\right)}{x^{2} + 3}\right)}{x^{2} + 3}$$

Simplify

The third derivative [src]

   /                           /         2 \       /         2 \                 \
   |               /        3\ |      4*x  |       |      2*x  | /             4\|
   |               \-1 + 6*x /*|-1 + ------|   2*x*|-1 + ------|*\5 - 2*x + 3*x /|
   |          3                |          2|       |          2|                 |
   |      18*x                 \     3 + x /       \     3 + x /                 |
12*|6*x - ------ + ------------------------- - ----------------------------------|
   |           2                  2                                2             |
   |      3 + x              3 + x                         /     2\              |
   \                                                       \3 + x /              /
----------------------------------------------------------------------------------
                                           2                                      
                                      3 + x

$$\frac{12 \left(- \frac{18 x^{3}}{x^{2} + 3} + 6 x - \frac{2 x \left(\frac{2 x^{2}}{x^{2} + 3} - 1\right) \left(3 x^{4} - 2 x + 5\right)}{\left(x^{2} + 3\right)^{2}} + \frac{\left(6 x^{3} - 1\right) \left(\frac{4 x^{2}}{x^{2} + 3} - 1\right)}{x^{2} + 3}\right)}{x^{2} + 3}$$

Simplify

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Derivative of (3x^4-2x+5)/(x^2+3)

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Piecewise:

The solution