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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $2 x + 2$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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Piecewise:

The solution