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Identical expressions
(z*z- two *z+ one)/(z*z- three *z+ two)
(z multiply by z minus 2 multiply by z plus 1) divide by (z multiply by z minus 3 multiply by z plus 2)
(z multiply by z minus two multiply by z plus one) divide by (z multiply by z minus three multiply by z plus two)
(zz-2z+1)/(zz-3z+2)
zz-2z+1/zz-3z+2
(z*z-2*z+1) divide by (z*z-3*z+2)
Similar expressions
(z*z-2*z+1)/(z*z-3*z-2)
(z*z-2*z-1)/(z*z-3*z+2)
(z*z-2*z+1)/(z*z+3*z+2)
(z*z+2*z+1)/(z*z-3*z+2)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d z} f{\left(z \right)}$ :
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: $2 z - 2$
To find $\frac{d}{d z} g{\left(z \right)}$ :
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: $2 z - 3$
Now plug in to the quotient rule:

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

$- \frac{1}{z^{2} - 4 z + 4}$

The answer is:

$- \frac{1}{z^{2} - 4 z + 4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

The graph:

Piecewise:

The solution

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How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution

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