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((x^ three)- three x^ two)/((x- one)^3)
((x cubed ) minus 3x squared ) divide by ((x minus 1) cubed )
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((x to the power of 3)-3x to the power of 2)/((x-1) to the power of 3)
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((x^3)-3x^2) divide by ((x-1)^3)
Similar expressions
((x^3)+3x^2)/((x-1)^3)
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The derivative of the function/ ((x^3)-3x^2)/((x-1)^3)

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

The solution

You have entered [src]

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

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

  / 3      2\
d |x  - 3*x |
--|---------|
dx|        3|
  \ (x - 1) /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{3} - 3 x^{2}$ term by term:
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  2. 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: $- 6 x$
  The result is: $3 x^{2} - 6 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 1$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
  1. Differentiate $x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $3 \left(x - 1\right)^{2}$
Now plug in to the quotient rule:

$\frac{\left(x - 1\right)^{3} \cdot \left(3 x^{2} - 6 x\right) - 3 \left(x - 1\right)^{2} \left(x^{3} - 3 x^{2}\right)}{\left(x - 1\right)^{6}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          2     / 3      2\
-6*x + 3*x    3*\x  - 3*x /
----------- - -------------
         3              4  
  (x - 1)        (x - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /        2                        \
   |     5*x *(-3 + x)   9*x*(-2 + x)|
12*|-4 - ------------- + ------------|
   |               3              2  |
   \       (-1 + x)       (-1 + x)   /
--------------------------------------
                      3               
              (-1 + x)

$$\frac{12 \left(- \frac{5 x^{2} \left(x - 3\right)}{\left(x - 1\right)^{3}} + \frac{9 x \left(x - 2\right)}{\left(x - 1\right)^{2}} - 4\right)}{\left(x - 1\right)^{3}}$$

Simplify

The graph

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

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The solution