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

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

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x - 1$ 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: $2$
  The result is: $2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $2 x - 2$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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