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Identical expressions
(x- three)^ two /(four (x- one))
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Similar expressions
(x-3)^2/(4(x+1))
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The derivative of the function/ (x-3)^2/(4(x-1))

Derivative of (x-3)^2/(4(x-1))

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(x - 3\right)^{2}$ and $g{\left(x \right)} = 4 x - 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x - 3$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 3\right)$ :
  1. Differentiate $x - 3$ term by term:
    1. The derivative of the constant $-3$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x - 6$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $4 x - 4$ 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: $4$
  The result is: $4$
Now plug in to the quotient rule:

$\frac{- 4 \left(x - 3\right)^{2} + \left(2 x - 6\right) \left(4 x - 4\right)}{\left(4 x - 4\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (x-3)^2/(4(x-1))

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Piecewise:

The solution