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Graphing y =:
(x^2+x-2)/(x-3)
Identical expressions
(x^ two +x- two)/(x- three)
(x squared plus x minus 2) divide by (x minus 3)
(x to the power of two plus x minus two) divide by (x minus three)
(x2+x-2)/(x-3)
x2+x-2/x-3
(x²+x-2)/(x-3)
(x to the power of 2+x-2)/(x-3)
x^2+x-2/x-3
(x^2+x-2) divide by (x-3)
Similar expressions
(x^2+x+2)/(x-3)
(x^2+x-2)/(x+3)
(x^2-x-2)/(x-3)

The derivative of the function/ (x^2+x-2)/(x-3)

Derivative of (x^2+x-2)/(x-3)

The solution

You have entered [src]

 2        
x  + x - 2
----------
  x - 3

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

(x^2 + x - 2)/(x - 3)

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 $g{\left(x \right)} = x - 3$ .

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 $x - 3$ term by term:
  1. The derivative of the constant $-3$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

$\frac{x^{2} - 6 x - 1}{x^{2} - 6 x + 9}$

The answer is:

$\frac{x^{2} - 6 x - 1}{x^{2} - 6 x + 9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           2        
1 + 2*x   x  + x - 2
------- - ----------
 x - 3            2 
           (x - 3)

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

Simplify

The second derivative [src]

  /              2          \
  |    -2 + x + x    1 + 2*x|
2*|1 + ----------- - -------|
  |             2     -3 + x|
  \     (-3 + x)            /
-----------------------------
            -3 + x

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

Simplify

3-я производная [src]

  /                         2\
  |     1 + 2*x   -2 + x + x |
6*|-1 + ------- - -----------|
  |      -3 + x            2 |
  \                (-3 + x)  /
------------------------------
                  2           
          (-3 + x)

$$\frac{6 \left(-1 + \frac{2 x + 1}{x - 3} - \frac{x^{2} + x - 2}{\left(x - 3\right)^{2}}\right)}{\left(x - 3\right)^{2}}$$

Simplify

The third derivative [src]

  /                         2\
  |     1 + 2*x   -2 + x + x |
6*|-1 + ------- - -----------|
  |      -3 + x            2 |
  \                (-3 + x)  /
------------------------------
                  2           
          (-3 + x)

$$\frac{6 \left(-1 + \frac{2 x + 1}{x - 3} - \frac{x^{2} + x - 2}{\left(x - 3\right)^{2}}\right)}{\left(x - 3\right)^{2}}$$

Simplify

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Derivative of (x^2+x-2)/(x-3)

The graph:

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The solution