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Graphing y =:
(x^2-x+2)/(x+1)
Identical expressions
(x^ two -x+ two)/(x+ one)
(x squared minus x plus 2) divide by (x plus 1)
(x to the power of two minus x plus two) divide by (x plus one)
(x2-x+2)/(x+1)
x2-x+2/x+1
(x²-x+2)/(x+1)
(x to the power of 2-x+2)/(x+1)
x^2-x+2/x+1
(x^2-x+2) divide by (x+1)
Similar expressions
(x^2+x+2)/(x+1)
(x^2-x-2)/(x+1)
(x^2-x+2)/(x-1)

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: $x^{2}$ goes to $2 x$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $-1$
  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