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Identical expressions
(x*(x- two))/(two *(x- one)^ two)
(x multiply by (x minus 2)) divide by (2 multiply by (x minus 1) squared )
(x multiply by (x minus two)) divide by (two multiply by (x minus one) to the power of two)
(x*(x-2))/(2*(x-1)2)
x*x-2/2*x-12
(x*(x-2))/(2*(x-1)²)
(x*(x-2))/(2*(x-1) to the power of 2)
(x(x-2))/(2(x-1)^2)
(x(x-2))/(2(x-1)2)
xx-2/2x-12
xx-2/2x-1^2
(x*(x-2)) divide by (2*(x-1)^2)
Similar expressions
(x*(x+2))/(2*(x-1)^2)
(x*(x-2))/(2*(x+1)^2)

The derivative of the function/ (x*(x-2))/(2*(x-1)^2)

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

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = x - 2$ ; 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: $1$
  The result is: $2 x - 2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = x - 1$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  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:
    
    $2 x - 2$
  So, the result is: $4 x - 4$
Now plug in to the quotient rule:

$\frac{- x \left(x - 2\right) \left(4 x - 4\right) + 2 \left(x - 1\right)^{2} \cdot \left(2 x - 2\right)}{4 \left(x - 1\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1            1                x*(4 - 4*x)*(x - 2)
x*---------- + ----------*(x - 2) + -------------------
           2            2                         4    
  2*(x - 1)    2*(x - 1)                 4*(x - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution

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