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Integral of d{x}:
x^3/(x-2)
How do you in partial fractions?:
x^3/(x-2)
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x^ three /(x- two)
x cubed divide by (x minus 2)
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Similar expressions
x^3/(x+2)

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

Derivative of x^3/(x-2)

The solution

You have entered [src]

   3 
  x  
-----
x - 2

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

x^3/(x - 2)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of x^3/(x-2)

The graph:

Piecewise:

The solution