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(zz-3z+2)/(zz-2z+1)
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The derivative of the function/ (z*z-3*z+2)/(z*z-2*z+1)

Derivative of (zz-3z+2)/(zz-2z+1)

The solution

You have entered [src]

z*z - 3*z + 2
-------------
z*z - 2*z + 1

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d z} f{\left(z \right)}$ :
1. Differentiate $z^{2} - 3 z + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. Apply the power rule: $z^{2}$ goes to $2 z$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $z$ goes to $1$
    So, the result is: $-3$
  The result is: $2 z - 3$
To find $\frac{d}{d z} g{\left(z \right)}$ :
1. Differentiate $z^{2} - 2 z + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $z^{2}$ goes to $2 z$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $z$ goes to $1$
    So, the result is: $-2$
  The result is: $2 z - 2$
Now plug in to the quotient rule:

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

$\frac{1}{z^{2} - 2 z + 1}$

The answer is:

$\frac{1}{z^{2} - 2 z + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /    /               2 \                                       \
  |    |     4*(-1 + z)  | /     2      \                        |
  |    |-1 + ------------|*\2 + z  - 3*z/                        |
  |    |          2      |                                       |
  |    \     1 + z  - 2*z/                  2*(-1 + z)*(-3 + 2*z)|
2*|1 + ---------------------------------- - ---------------------|
  |                    2                              2          |
  \               1 + z  - 2*z                   1 + z  - 2*z    /
------------------------------------------------------------------
                                2                                 
                           1 + z  - 2*z

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

Simplify

The third derivative [src]

  /                                                      /               2 \               \
  |                                                      |     2*(-1 + z)  | /     2      \|
  |                                           4*(-1 + z)*|-1 + ------------|*\2 + z  - 3*z/|
  |          /               2 \                         |          2      |               |
  |          |     4*(-1 + z)  |                         \     1 + z  - 2*z/               |
6*|2 - 2*z + |-1 + ------------|*(-3 + 2*z) - ---------------------------------------------|
  |          |          2      |                                    2                      |
  \          \     1 + z  - 2*z/                               1 + z  - 2*z                /
--------------------------------------------------------------------------------------------
                                                    2                                       
                                      /     2      \                                        
                                      \1 + z  - 2*z/

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

Simplify

The graph

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Derivative of (zz-3z+2)/(zz-2z+1)

The graph:

Piecewise:

The solution

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Identical expressions

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Derivative of (z*z-3*z+2)/(z*z-2*z+1)

The graph:

Piecewise:

The solution

Derivative of (zz-3z+2)/(zz-2z+1)