Derivative of (3x-1)/((x+2)(x-3))

The solution

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

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 x - 1$ and $g{\left(x \right)} = \left(x - 3\right) \left(x + 2\right)$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. 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: $3$
  The result is: $3$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x - 3$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  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: $1$
  $g{\left(x \right)} = x + 2$ ; 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: $1$
  The result is: $2 x - 1$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                      /     -1 + 2*x   -1 + 2*x              /  1        1  \\
6 - 12*x + (-1 + 3*x)*|-2 + -------- + -------- + (-1 + 2*x)*|------ + -----||
                      \      -3 + x     2 + x                \-3 + x   2 + x//
------------------------------------------------------------------------------
                                      2        2                              
                              (-3 + x) *(2 + x)

$$\frac{- 12 x + \left(3 x - 1\right) \left(\left(2 x - 1\right) \left(\frac{1}{x + 2} + \frac{1}{x - 3}\right) - 2 + \frac{2 x - 1}{x + 2} + \frac{2 x - 1}{x - 3}\right) + 6}{\left(x - 3\right)^{2} \left(x + 2\right)^{2}}$$

Simplify

The third derivative [src]

                 /                                                                                                                     /  1        1  \              /  1        1  \                   \                                                              
                 |                                                                                                          (-1 + 2*x)*|------ + -----|   (-1 + 2*x)*|------ + -----|                   |                                                              
                 |    8        8                  /    1          1              1        \   3*(-1 + 2*x)   3*(-1 + 2*x)              \-3 + x   2 + x/              \-3 + x   2 + x/     4*(-1 + 2*x)  |   9*(-1 + 2*x)   9*(-1 + 2*x)                /  1        1  \
-18 - (-1 + 3*x)*|- ------ - ----- + 2*(-1 + 2*x)*|--------- + -------- + ----------------| + ------------ + ------------ + --------------------------- + --------------------------- + ----------------| + ------------ + ------------ + 9*(-1 + 2*x)*|------ + -----|
                 |  -3 + x   2 + x                |        2          2   (-3 + x)*(2 + x)|            2              2                -3 + x                        2 + x              (-3 + x)*(2 + x)|      -3 + x         2 + x                    \-3 + x   2 + x/
                 \                                \(-3 + x)    (2 + x)                    /    (-3 + x)        (2 + x)                                                                                  /                                                              
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                   2        2                                                                                                                          
                                                                                                                           (-3 + x) *(2 + x)

$$\frac{9 \left(2 x - 1\right) \left(\frac{1}{x + 2} + \frac{1}{x - 3}\right) - \left(3 x - 1\right) \left(2 \left(2 x - 1\right) \left(\frac{1}{\left(x + 2\right)^{2}} + \frac{1}{\left(x - 3\right) \left(x + 2\right)} + \frac{1}{\left(x - 3\right)^{2}}\right) + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 2} + \frac{1}{x - 3}\right)}{x + 2} - \frac{8}{x + 2} + \frac{3 \left(2 x - 1\right)}{\left(x + 2\right)^{2}} + \frac{\left(2 x - 1\right) \left(\frac{1}{x + 2} + \frac{1}{x - 3}\right)}{x - 3} - \frac{8}{x - 3} + \frac{4 \left(2 x - 1\right)}{\left(x - 3\right) \left(x + 2\right)} + \frac{3 \left(2 x - 1\right)}{\left(x - 3\right)^{2}}\right) - 18 + \frac{9 \left(2 x - 1\right)}{x + 2} + \frac{9 \left(2 x - 1\right)}{x - 3}}{\left(x - 3\right)^{2} \left(x + 2\right)^{2}}$$

Simplify

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Derivative of (3x-1)/((x+2)(x-3))

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