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Identical expressions
(x- one)/((x+ two)sqrt(x- two))
(x minus 1) divide by ((x plus 2) square root of (x minus 2))
(x minus one) divide by ((x plus two) square root of (x minus two))
(x-1)/((x+2)√(x-2))
x-1/x+2sqrtx-2
(x-1) divide by ((x+2)sqrt(x-2))
Similar expressions
(x-1)/((x+2)sqrt(x+2))
(x-1)/((x-2)sqrt(x-2))
(x+1)/((x+2)sqrt(x-2))

The derivative of the function/ (x-1)/((x+2)sqrt(x-2))

Derivative of (x-1)/((x+2)sqrt(x-2))

The solution

You have entered [src]

      x - 1      
-----------------
          _______
(x + 2)*\/ x - 2

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

(x - 1)/(((x + 2)*sqrt(x - 2)))

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
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: $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sqrt{x - 2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x - 2$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
    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 of the chain rule is:
    
    $\frac{1}{2 \sqrt{x - 2}}$
  $g{\left(x \right)} = x + 2$ ; 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: $\sqrt{x - 2} + \frac{x + 2}{2 \sqrt{x - 2}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

           /                                      /    ________     2 + x   \   /    ________     2 + x   \ /    3          8              4        \   /  1        2  \ /    ________     2 + x   \                                                           /    ________     2 + x   \      /    ________     2 + x   \     /  1        2  \ /    ________     2 + x   \\                      /    ________     2 + x   \     /  1        2  \ /    ________     2 + x   \      /    ________     2 + x   \
           |    /    2 + x \     /    2 + x \   4*|2*\/ -2 + x  + ----------|   |2*\/ -2 + x  + ----------|*|--------- + -------- + ----------------|   |------ + -----|*|2*\/ -2 + x  + ----------|   /    2 + x \ /  1        2  \        /    2 + x \     8*|2*\/ -2 + x  + ----------|   12*|2*\/ -2 + x  + ----------|   2*|------ + -----|*|2*\/ -2 + x  + ----------||     /    2 + x \   6*|2*\/ -2 + x  + ----------|   6*|------ + -----|*|2*\/ -2 + x  + ----------|   12*|2*\/ -2 + x  + ----------|
           |  3*|2 - ------|   3*|4 - ------|     |                 ________|   |                 ________| |        2          2   (-2 + x)*(2 + x)|   \-2 + x   2 + x/ |                 ________|   |4 - ------|*|------ + -----|      6*|4 - ------|       |                 ________|      |                 ________|     \-2 + x   2 + x/ |                 ________||   6*|4 - ------|     |                 ________|     \-2 + x   2 + x/ |                 ________|      |                 ________|
           |    \    -2 + x/     \    -2 + x/     \               \/ -2 + x /   \               \/ -2 + x / \(-2 + x)    (2 + x)                    /                    \               \/ -2 + x /   \    -2 + x/ \-2 + x   2 + x/        \    -2 + x/       \               \/ -2 + x /      \               \/ -2 + x /                      \               \/ -2 + x /|     \    -2 + x/     \               \/ -2 + x /                      \               \/ -2 + x /      \               \/ -2 + x /
- (-1 + x)*|- -------------- - -------------- + ----------------------------- + --------------------------------------------------------------------- + -------------------------------------------- - ----------------------------- - ------------------- + ----------------------------- + ------------------------------ + ----------------------------------------------| - -------------- + ----------------------------- + ---------------------------------------------- + ------------------------------
           |           5/2              5/2                       3                                             -2 + x                                                           2                                      3/2                    3/2                         2                                       2                         (-2 + x)*(2 + x)               |            3/2                       2                                 -2 + x                              (-2 + x)*(2 + x)       
           \   (-2 + x)         (-2 + x)                  (-2 + x)                                                                                                       (-2 + x)                               (-2 + x)               (-2 + x)   *(2 + x)         (-2 + x) *(2 + x)               (-2 + x)*(2 + x)                                                         /    (-2 + x)                  (-2 + x)                                                                                             
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                                                                                                                                                                                                                                                            2                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                   8*(2 + x)

$$\frac{- \frac{6 \left(4 - \frac{x + 2}{x - 2}\right)}{\left(x - 2\right)^{\frac{3}{2}}} - \left(x - 1\right) \left(- \frac{3 \left(2 - \frac{x + 2}{x - 2}\right)}{\left(x - 2\right)^{\frac{5}{2}}} - \frac{\left(4 - \frac{x + 2}{x - 2}\right) \left(\frac{2}{x + 2} + \frac{1}{x - 2}\right)}{\left(x - 2\right)^{\frac{3}{2}}} - \frac{6 \left(4 - \frac{x + 2}{x - 2}\right)}{\left(x - 2\right)^{\frac{3}{2}} \left(x + 2\right)} - \frac{3 \left(4 - \frac{x + 2}{x - 2}\right)}{\left(x - 2\right)^{\frac{5}{2}}} + \frac{\left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right) \left(\frac{8}{\left(x + 2\right)^{2}} + \frac{4}{\left(x - 2\right) \left(x + 2\right)} + \frac{3}{\left(x - 2\right)^{2}}\right)}{x - 2} + \frac{2 \left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right) \left(\frac{2}{x + 2} + \frac{1}{x - 2}\right)}{\left(x - 2\right) \left(x + 2\right)} + \frac{12 \left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right)}{\left(x - 2\right) \left(x + 2\right)^{2}} + \frac{\left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right) \left(\frac{2}{x + 2} + \frac{1}{x - 2}\right)}{\left(x - 2\right)^{2}} + \frac{8 \left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right)}{\left(x - 2\right)^{2} \left(x + 2\right)} + \frac{4 \left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right)}{\left(x - 2\right)^{3}}\right) + \frac{6 \left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right) \left(\frac{2}{x + 2} + \frac{1}{x - 2}\right)}{x - 2} + \frac{12 \left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right)}{\left(x - 2\right) \left(x + 2\right)} + \frac{6 \left(2 \sqrt{x - 2} + \frac{x + 2}{\sqrt{x - 2}}\right)}{\left(x - 2\right)^{2}}}{8 \left(x + 2\right)^{2}}$$

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Derivative of (x-1)/((x+2)sqrt(x-2))

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The solution