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Identical expressions
x(x- eighteen)/(x- nine)^ two
x(x minus 18) divide by (x minus 9) squared
x(x minus eighteen) divide by (x minus nine) to the power of two
x(x-18)/(x-9)2
xx-18/x-92
x(x-18)/(x-9)²
x(x-18)/(x-9) to the power of 2
xx-18/x-9^2
x(x-18) divide by (x-9)^2
Similar expressions
x(x+18)/(x-9)^2
x(x-18)/(x+9)^2

The derivative of the function/ x(x-18)/(x-9)^2

Derivative of x(x-18)/(x-9)^2

The solution

You have entered [src]

x*(x - 18)
----------
        2 
 (x - 9)

$$\frac{x \left(x - 18\right)}{\left(x - 9\right)^{2}}$$

(x*(x - 18))/(x - 9)^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x \left(x - 18\right)$ and $g{\left(x \right)} = \left(x - 9\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = x - 18$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 18$ term by term:
    1. The derivative of the constant $-18$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result is: $2 x - 18$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 9$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 9\right)$ :
  1. Differentiate $x - 9$ term by term:
    1. The derivative of the constant $-9$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x - 18$
Now plug in to the quotient rule:

$\frac{- x \left(x - 18\right) \left(2 x - 18\right) + \left(x - 9\right)^{2} \left(2 x - 18\right)}{\left(x - 9\right)^{4}}$
Now simplify:

$\frac{162}{x^{3} - 27 x^{2} + 243 x - 729}$

The answer is:

$\frac{162}{x^{3} - 27 x^{2} + 243 x - 729}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-18 + 2*x   x*(18 - 2*x)*(x - 18)
--------- + ---------------------
        2                 4      
 (x - 9)           (x - 9)

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

Simplify

The second derivative [src]

  /     x*(-18 + x)\
6*|-1 + -----------|
  |              2 |
  \      (-9 + x)  /
--------------------
             2      
     (-9 + x)

$$\frac{6 \left(\frac{x \left(x - 18\right)}{\left(x - 9\right)^{2}} - 1\right)}{\left(x - 9\right)^{2}}$$

Simplify

The third derivative [src]

   /    x*(-18 + x)\
24*|1 - -----------|
   |             2 |
   \     (-9 + x)  /
--------------------
             3      
     (-9 + x)

$$\frac{24 \left(- \frac{x \left(x - 18\right)}{\left(x - 9\right)^{2}} + 1\right)}{\left(x - 9\right)^{3}}$$

Simplify

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Derivative of x(x-18)/(x-9)^2

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The solution