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The derivative of the function/ (x-5)/(x-3)

Derivative of (x-5)/(x-3)

The solution

You have entered [src]

x - 5
-----
x - 3

\frac{x - 5}{x - 3}

(x - 5)/(x - 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x - 5$ and $g{\left(x \right)} = x - 3$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x - 5$ term by term:
  1. The derivative of the constant $-5$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1      x - 5  
----- - --------
x - 3          2
        (x - 3)

- \frac{x - 5}{\left(x - 3\right)^{2}} + \frac{1}{x - 3}

Simplify

The second derivative [src]

  /     -5 + x\
2*|-1 + ------|
  \     -3 + x/
---------------
           2   
   (-3 + x)

\frac{2 \left(\frac{x - 5}{x - 3} - 1\right)}{\left(x - 3\right)^{2}}

Simplify

The third derivative [src]

  /    -5 + x\
6*|1 - ------|
  \    -3 + x/
--------------
          3   
  (-3 + x)

\frac{6 \left(- \frac{x - 5}{x - 3} + 1\right)}{\left(x - 3\right)^{3}}

Simplify