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Derivative of cos(x/3)
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Graphing y =:
(x^2-6x+13)/(x-3)
Identical expressions
(x^ two -6x+ thirteen)/(x- three)
(x squared minus 6x plus 13) divide by (x minus 3)
(x to the power of two minus 6x plus thirteen) divide by (x minus three)
(x2-6x+13)/(x-3)
x2-6x+13/x-3
(x²-6x+13)/(x-3)
(x to the power of 2-6x+13)/(x-3)
x^2-6x+13/x-3
(x^2-6x+13) divide by (x-3)
Similar expressions
(x^2+6x+13)/(x-3)
(x^2-6x-13)/(x-3)
(x^2-6x+13)/(x+3)

The derivative of the function/ (x^2-6x+13)/(x-3)

Derivative of (x^2-6x+13)/(x-3)

The solution

You have entered [src]

 2           
x  - 6*x + 13
-------------
    x - 3

$$\frac{\left(x^{2} - 6 x\right) + 13}{x - 3}$$

(x^2 - 6*x + 13)/(x - 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} - 6 x + 13$ and $g{\left(x \right)} = x - 3$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} - 6 x + 13$ term by term:
  1. The derivative of the constant $13$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  3. 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: $-6$
  The result is: $2 x - 6$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x - 3$ term by term:
  1. The derivative of the constant $-3$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{- x^{2} + 6 x + \left(x - 3\right) \left(2 x - 6\right) - 13}{\left(x - 3\right)^{2}}$
Now simplify:

$\frac{x^{2} - 6 x + 5}{x^{2} - 6 x + 9}$

The answer is:

$\frac{x^{2} - 6 x + 5}{x^{2} - 6 x + 9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            2           
-6 + 2*x   x  - 6*x + 13
-------- - -------------
 x - 3               2  
              (x - 3)

$$\frac{2 x - 6}{x - 3} - \frac{\left(x^{2} - 6 x\right) + 13}{\left(x - 3\right)^{2}}$$

Simplify

The second derivative [src]

  /           2      \
  |     13 + x  - 6*x|
2*|-1 + -------------|
  |               2  |
  \       (-3 + x)   /
----------------------
        -3 + x

$$\frac{2 \left(-1 + \frac{x^{2} - 6 x + 13}{\left(x - 3\right)^{2}}\right)}{x - 3}$$

Simplify

The third derivative [src]

  /          2      \
  |    13 + x  - 6*x|
6*|1 - -------------|
  |              2  |
  \      (-3 + x)   /
---------------------
              2      
      (-3 + x)

$$\frac{6 \left(1 - \frac{x^{2} - 6 x + 13}{\left(x - 3\right)^{2}}\right)}{\left(x - 3\right)^{2}}$$

Simplify

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Derivative of (x^2-6x+13)/(x-3)

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The solution