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Identical expressions
x*((four x- one)/(2x-4))
x multiply by ((4x minus 1) divide by (2x minus 4))
x multiply by ((four x minus one) divide by (2x minus 4))
x((4x-1)/(2x-4))
x4x-1/2x-4
x*((4x-1) divide by (2x-4))
Similar expressions
x*((4x-1)/(2x+4))
x*((4x+1)/(2x-4))

The derivative of the function/ x*((4x-1)/(2x-4))

Derivative of x*((4x-1)/(2x-4))

The solution

You have entered [src]

  4*x - 1
x*-------
  2*x - 4

$$x \frac{4 x - 1}{2 x - 4}$$

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x \left(4 x - 1\right)$ and $g{\left(x \right)} = 2 x - 4$ .

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 $1$
  $g{\left(x \right)} = 4 x - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $4 x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $4$
    The result is: $4$
  The result is: $8 x - 1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 x - 4$ term by term:
  1. The derivative of the constant $-4$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $2$
  The result is: $2$
Now plug in to the quotient rule:

$\frac{- 2 x \left(4 x - 1\right) + \left(2 x - 4\right) \left(8 x - 1\right)}{\left(2 x - 4\right)^{2}}$
Now simplify:

$\frac{2 x^{2} - 8 x + 1}{x^{2} - 4 x + 4}$

The answer is:

$\frac{2 x^{2} - 8 x + 1}{x^{2} - 4 x + 4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                 /    -1 + 4*x\
               x*|4 - --------|
    -1 + 4*x     \     -2 + x /
4 - -------- - ----------------
     -2 + x         -2 + x     
-------------------------------
             -2 + x

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of x*((4x-1)/(2x-4))

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The solution