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Identical expressions
(x^ two - three)/(x^ three + two)
(x squared minus 3) divide by (x cubed plus 2)
(x to the power of two minus three) divide by (x to the power of three plus two)
(x2-3)/(x3+2)
x2-3/x3+2
(x²-3)/(x³+2)
(x to the power of 2-3)/(x to the power of 3+2)
x^2-3/x^3+2
(x^2-3) divide by (x^3+2)
Similar expressions
(x^2-3)/(x^3-2)
(x^2+3)/(x^3+2)

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

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

The solution

You have entered [src]

 2    
x  - 3
------
 3    
x  + 2

$$\frac{x^{2} - 3}{x^{3} + 2}$$

  / 2    \
d |x  - 3|
--|------|
dx| 3    |
  \x  + 2/

$$\frac{d}{d x} \frac{x^{2} - 3}{x^{3} + 2}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            2 / 2    \
 2*x     3*x *\x  - 3/
------ - -------------
 3                 2  
x  + 2     / 3    \   
           \x  + 2/

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

Simplify

The second derivative [src]

  /                 /         3 \          \
  |                 |      3*x  | /      2\|
  |             3*x*|-1 + ------|*\-3 + x /|
  |        3        |          3|          |
  |     6*x         \     2 + x /          |
2*|1 - ------ + ---------------------------|
  |         3                   3          |
  \    2 + x               2 + x           /
--------------------------------------------
                        3                   
                   2 + x

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

Simplify

The third derivative [src]

  /                   /        3          6  \        /         3 \\
  |     2   /      2\ |    18*x       27*x   |      2 |      3*x  ||
6*|- 3*x  - \-3 + x /*|1 - ------ + ---------| + 6*x *|-1 + ------||
  |                   |         3           2|        |          3||
  |                   |    2 + x    /     3\ |        \     2 + x /|
  \                   \             \2 + x / /                     /
--------------------------------------------------------------------
                                     2                              
                             /     3\                               
                             \2 + x /

$$\frac{6 \cdot \left(6 x^{2} \cdot \left(\frac{3 x^{3}}{x^{3} + 2} - 1\right) - 3 x^{2} - \left(x^{2} - 3\right) \left(\frac{27 x^{6}}{\left(x^{3} + 2\right)^{2}} - \frac{18 x^{3}}{x^{3} + 2} + 1\right)\right)}{\left(x^{3} + 2\right)^{2}}$$

Simplify

The graph

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

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The solution