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Identical expressions
((x- three)^ two *(two x+ three))/((x- one)^ three *(x-2))
((x minus 3) squared multiply by (2x plus 3)) divide by ((x minus 1) cubed multiply by (x minus 2))
((x minus three) to the power of two multiply by (two x plus three)) divide by ((x minus one) to the power of three multiply by (x minus 2))
((x-3)2*(2x+3))/((x-1)3*(x-2))
x-32*2x+3/x-13*x-2
((x-3)²*(2x+3))/((x-1)³*(x-2))
((x-3) to the power of 2*(2x+3))/((x-1) to the power of 3*(x-2))
((x-3)^2(2x+3))/((x-1)^3(x-2))
((x-3)2(2x+3))/((x-1)3(x-2))
x-322x+3/x-13x-2
x-3^22x+3/x-1^3x-2
((x-3)^2*(2x+3)) divide by ((x-1)^3*(x-2))
Similar expressions
((x-3)^2*(2x+3))/((x-1)^3*(x+2))
((x-3)^2*(2x+3))/((x+1)^3*(x-2))
((x-3)^2*(2x-3))/((x-1)^3*(x-2))
((x+3)^2*(2x+3))/((x-1)^3*(x-2))

The derivative of the function/ ((x-3)^2*(2x+3))/((x-1)^3*(x-2))

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

The solution

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \left(x - 3\right)^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x - 3$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 3\right)$ :
    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: $1$
    The result of the chain rule is:
    
    $2 x - 6$
  $g{\left(x \right)} = 2 x + 3$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $2 x + 3$ term by term:
    1. The derivative of the constant $3$ 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: $2$
    The result is: $2$
  The result is: $2 \left(x - 3\right)^{2} + \left(2 x - 6\right) \left(2 x + 3\right)$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \left(x - 1\right)^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x - 1$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
    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: $1$
    The result of the chain rule is:
    
    $3 \left(x - 1\right)^{2}$
  $g{\left(x \right)} = x - 2$ ; 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: $3 \left(x - 2\right) \left(x - 1\right)^{2} + \left(x - 1\right)^{3}$
Now plug in to the quotient rule:

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

$\frac{\left(x - 3\right) \left(6 x \left(x - 2\right) \left(x - 1\right) - \left(x - 3\right) \left(2 x + 3\right) \left(4 x - 7\right)\right)}{\left(x - 2\right)^{2} \left(x - 1\right)^{4}}$

The answer is:

$\frac{\left(x - 3\right) \left(6 x \left(x - 2\right) \left(x - 1\right) - \left(x - 3\right) \left(2 x + 3\right) \left(4 x - 7\right)\right)}{\left(x - 2\right)^{2} \left(x - 1\right)^{4}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                     2           /-7 + 4*x              /  1        3   \   6*(-3 + 2*x)   3*(-7 + 4*x)\                           
             (-3 + x) *(3 + 2*x)*|-------- + (-7 + 4*x)*|------ + ------| - ------------ + ------------|                           
                                 \ -2 + x               \-2 + x   -1 + x/      -1 + x         -1 + x   /   12*x*(-7 + 4*x)*(-3 + x)
-18 + 12*x + ------------------------------------------------------------------------------------------- - ------------------------
                                                  (-1 + x)*(-2 + x)                                           (-1 + x)*(-2 + x)    
-----------------------------------------------------------------------------------------------------------------------------------
                                                                 3                                                                 
                                                         (-1 + x) *(-2 + x)

$$\frac{- \frac{12 x \left(x - 3\right) \left(4 x - 7\right)}{\left(x - 2\right) \left(x - 1\right)} + 12 x + \frac{\left(x - 3\right)^{2} \left(2 x + 3\right) \left(\left(4 x - 7\right) \left(\frac{3}{x - 1} + \frac{1}{x - 2}\right) - \frac{6 \left(2 x - 3\right)}{x - 1} + \frac{3 \left(4 x - 7\right)}{x - 1} + \frac{4 x - 7}{x - 2}\right)}{\left(x - 2\right) \left(x - 1\right)} - 18}{\left(x - 2\right) \left(x - 1\right)^{3}}$$

Simplify

The third derivative [src]

                                                    /                                                                                                                                      /  1        3   \                                    /  1        3   \                /  1        3   \                    \                                                                                        
                                                    |                                                                                                                           (-7 + 4*x)*|------ + ------|                       6*(-3 + 2*x)*|------ + ------|   3*(-7 + 4*x)*|------ + ------|                    |                                                                                        
                                        2           |  54*(-3 + 2*x)                /    1           6               3        \   3*(-7 + 4*x)   6*(-5 + 4*x)   21*(-7 + 4*x)              \-2 + x   -1 + x/     18*(-3 + 2*x)                  \-2 + x   -1 + x/                \-2 + x   -1 + x/     12*(-7 + 4*x)  |                                                                                        
                                (-3 + x) *(3 + 2*x)*|- ------------- + 2*(-7 + 4*x)*|--------- + --------- + -----------------| + ------------ + ------------ + ------------- + ---------------------------- - ----------------- - ------------------------------ + ------------------------------ + -----------------|                 /-7 + 4*x              /  1        3   \   6*(-3 + 2*x)   3*(-7 + 4*x)\
                                                    |            2                  |        2           2   (-1 + x)*(-2 + x)|            2              2               2                -2 + x              (-1 + x)*(-2 + x)               -1 + x                           -1 + x               (-1 + x)*(-2 + x)|   18*x*(-3 + x)*|-------- + (-7 + 4*x)*|------ + ------| - ------------ + ------------|
     18*(-7 + 4*x)*(-3 + 2*x)                       \    (-1 + x)                   \(-2 + x)    (-1 + x)                     /    (-2 + x)       (-1 + x)        (-1 + x)                                                                                                                                            /                 \ -2 + x               \-2 + x   -1 + x/      -1 + x         -1 + x   /
12 - ------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
        (-1 + x)*(-2 + x)                                                                                                                                          (-1 + x)*(-2 + x)                                                                                                                                                                        (-1 + x)*(-2 + x)                                  
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                       3                                                                                                                                                                                                       
                                                                                                                                                                                               (-1 + x) *(-2 + x)

$$\frac{\frac{18 x \left(x - 3\right) \left(\left(4 x - 7\right) \left(\frac{3}{x - 1} + \frac{1}{x - 2}\right) - \frac{6 \left(2 x - 3\right)}{x - 1} + \frac{3 \left(4 x - 7\right)}{x - 1} + \frac{4 x - 7}{x - 2}\right)}{\left(x - 2\right) \left(x - 1\right)} - \frac{\left(x - 3\right)^{2} \left(2 x + 3\right) \left(2 \left(4 x - 7\right) \left(\frac{6}{\left(x - 1\right)^{2}} + \frac{3}{\left(x - 2\right) \left(x - 1\right)} + \frac{1}{\left(x - 2\right)^{2}}\right) - \frac{6 \left(2 x - 3\right) \left(\frac{3}{x - 1} + \frac{1}{x - 2}\right)}{x - 1} + \frac{3 \left(4 x - 7\right) \left(\frac{3}{x - 1} + \frac{1}{x - 2}\right)}{x - 1} - \frac{54 \left(2 x - 3\right)}{\left(x - 1\right)^{2}} + \frac{21 \left(4 x - 7\right)}{\left(x - 1\right)^{2}} + \frac{6 \left(4 x - 5\right)}{\left(x - 1\right)^{2}} + \frac{\left(4 x - 7\right) \left(\frac{3}{x - 1} + \frac{1}{x - 2}\right)}{x - 2} - \frac{18 \left(2 x - 3\right)}{\left(x - 2\right) \left(x - 1\right)} + \frac{12 \left(4 x - 7\right)}{\left(x - 2\right) \left(x - 1\right)} + \frac{3 \left(4 x - 7\right)}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right) \left(x - 1\right)} + 12 - \frac{18 \left(2 x - 3\right) \left(4 x - 7\right)}{\left(x - 2\right) \left(x - 1\right)}}{\left(x - 2\right) \left(x - 1\right)^{3}}$$

Simplify

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

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The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution

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