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Derivative of (x^2-4)^3
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Derivative of (x^2+3x)(2x-4)
Graphing y =:
((x-2)/(x+1))^2
Identical expressions
((x- two)/(x+ one))^ two
((x minus 2) divide by (x plus 1)) squared
((x minus two) divide by (x plus one)) to the power of two
((x-2)/(x+1))2
x-2/x+12
((x-2)/(x+1))²
((x-2)/(x+1)) to the power of 2
x-2/x+1^2
((x-2) divide by (x+1))^2
Similar expressions
((x-2)/(x-1))^2
((x+2)/(x+1))^2

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

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

The solution

You have entered [src]

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

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

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

Detail solution

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

$\frac{6 \left(x - 2\right)}{\left(x + 1\right) \left(x + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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The solution