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$-((z)\((z-2)^2))$

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Derivative of:
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Derivative of ex^2
Identical expressions
-((z)\((z- two)^ two))
minus ((z)\((z minus 2) squared ))
minus ((z)\((z minus two) to the power of two))
-((z)\((z-2)2))
-z\z-22
-((z)\((z-2)²))
-((z)\((z-2) to the power of 2))
-z\z-2^2
Similar expressions
((z)\((z-2)^2))
-((z)\((z+2)^2))

The derivative of the function/ -((z)\((z-2)^2))

Derivative of -((z)\((z-2)^2))

The solution

You have entered [src]

  -z    
--------
       2
(z - 2)

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

-z/(z - 2)^2

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the quotient rule, which is:
  
  $\frac{d}{d z} \frac{f{\left(z \right)}}{g{\left(z \right)}} = \frac{- f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}}{g^{2}{\left(z \right)}}$
  
  $f{\left(z \right)} = z$ and $g{\left(z \right)} = \left(z - 2\right)^{2}$ .
  
  To find $\frac{d}{d z} f{\left(z \right)}$ :
  1. Apply the power rule: $z$ goes to $1$
  To find $\frac{d}{d z} g{\left(z \right)}$ :
  1. Let $u = z - 2$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d z} \left(z - 2\right)$ :
    1. Differentiate $z - 2$ term by term:
      1. The derivative of the constant $-2$ is zero.
      2. Apply the power rule: $z$ goes to $1$
      The result is: $1$
    The result of the chain rule is:
    
    $2 z - 4$
  Now plug in to the quotient rule:
  
  $\frac{- z \left(2 z - 4\right) + \left(z - 2\right)^{2}}{\left(z - 2\right)^{4}}$
So, the result is: $- \frac{- z \left(2 z - 4\right) + \left(z - 2\right)^{2}}{\left(z - 2\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     1       z*(4 - 2*z)
- -------- - -----------
         2            4 
  (z - 2)      (z - 2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

$Derivative of -((z)\((z-2)^2))$