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(x^ three - one)/(x+ two)
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Similar expressions
(x^3+1)/(x+2)
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The derivative of the function/ (x^3-1)/(x+2)

Derivative of (x^3-1)/(x+2)

The solution

You have entered [src]

 3    
x  - 1
------
x + 2

$$\frac{x^{3} - 1}{x + 2}$$

(x^3 - 1)/(x + 2)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    3            2
   x  - 1     3*x 
- -------- + -----
         2   x + 2
  (x + 2)

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

Simplify

The second derivative [src]

  /            3        2\
  |      -1 + x      3*x |
2*|3*x + -------- - -----|
  |             2   2 + x|
  \      (2 + x)         /
--------------------------
          2 + x

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (x^3-1)/(x+2)

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The solution