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

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

The solution

You have entered [src]

       2
(x - 2) 
--------
 x + 1

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x - 2$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
  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:
  
  $2 x - 4$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /             2             \
  |     (-2 + x)    2*(-2 + x)|
6*|-1 - --------- + ----------|
  |             2     1 + x   |
  \      (1 + x)              /
-------------------------------
                   2           
            (1 + x)

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

Simplify

The graph

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

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The solution