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Graphing y =:
(x^2-9*x)/(x^2-3*x)
Limit of the function:
(x^2-9*x)/(x^2-3*x)
Identical expressions
(x^ two - nine *x)/(x^ two - three *x)
(x squared minus 9 multiply by x) divide by (x squared minus 3 multiply by x)
(x to the power of two minus nine multiply by x) divide by (x to the power of two minus three multiply by x)
(x2-9*x)/(x2-3*x)
x2-9*x/x2-3*x
(x²-9*x)/(x²-3*x)
(x to the power of 2-9*x)/(x to the power of 2-3*x)
(x^2-9x)/(x^2-3x)
(x2-9x)/(x2-3x)
x2-9x/x2-3x
x^2-9x/x^2-3x
(x^2-9*x) divide by (x^2-3*x)
Similar expressions
(x^2+9*x)/(x^2-3*x)
(x^2-9*x)/(x^2+3*x)

The derivative of the function/ (x^2-9*x)/(x^2-3*x)

Derivative of (x^2-9x)/(x^2-3x)

The solution

You have entered [src]

 2      
x  - 9*x
--------
 2      
x  - 3*x

$$\frac{x^{2} - 9 x}{x^{2} - 3 x}$$

  / 2      \
d |x  - 9*x|
--|--------|
dx| 2      |
  \x  - 3*x/

$$\frac{d}{d x} \frac{x^{2} - 9 x}{x^{2} - 3 x}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} - 9 x$ and $g{\left(x \right)} = x^{2} - 3 x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} - 9 x$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  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: $-9$
  The result is: $2 x - 9$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} - 3 x$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  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: $-3$
  The result is: $2 x - 3$
Now plug in to the quotient rule:

$\frac{\left(2 x - 9\right) \left(x^{2} - 3 x\right) - \left(2 x - 3\right) \left(x^{2} - 9 x\right)}{\left(x^{2} - 3 x\right)^{2}}$
Now simplify:

$\frac{6}{x^{2} - 6 x + 9}$

The answer is:

$\frac{6}{x^{2} - 6 x + 9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                     / 2      \
-9 + 2*x   (3 - 2*x)*\x  - 9*x/
-------- + --------------------
 2                       2     
x  - 3*x       / 2      \      
               \x  - 3*x/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                             /              2\\
  |                                                             |    (-3 + 2*x) ||
  |          /              2\              (-9 + x)*(-3 + 2*x)*|2 - -----------||
  |          |    (-3 + 2*x) |                                  \     x*(-3 + x)/|
6*|3 - 2*x - |1 - -----------|*(-9 + 2*x) + -------------------------------------|
  \          \     x*(-3 + x)/                              -3 + x               /
----------------------------------------------------------------------------------
                                    2         2                                   
                                   x *(-3 + x)

$$\frac{6 \left(- \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \left(2 x - 9\right) + \frac{\left(2 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \left(x - 9\right) \left(2 x - 3\right)}{x - 3} - 2 x + 3\right)}{x^{2} \left(x - 3\right)^{2}}$$

Simplify

The graph

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Derivative of (x^2-9x)/(x^2-3x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (x^2-9*x)/(x^2-3*x)

The graph:

Piecewise:

The solution

Derivative of (x^2-9x)/(x^2-3x)