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

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

$\frac{x^{2} + 4 x + 3}{x^{2} + 4 x + 4}$

The answer is:

$\frac{x^{2} + 4 x + 3}{x^{2} + 4 x + 4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

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

The graph:

Piecewise:

The solution