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

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

The solution

You have entered [src]

 2        
x  + x + 2
----------
  x - 1

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + x + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  3. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x + 1$
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:

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

$\frac{x^{2} - 2 x - 3}{x^{2} - 2 x + 1}$

The answer is:

$\frac{x^{2} - 2 x - 3}{x^{2} - 2 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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The solution