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The derivative of the function/ y=(2x-1)/(x-3)

Derivative of y=(2x-1)/(x-3)

The solution

You have entered [src]

2*x - 1
-------
 x - 3

$$\frac{2 x - 1}{x - 3}$$

(2*x - 1)/(x - 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x - 1$ and $g{\left(x \right)} = x - 3$ .

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  2     2*x - 1 
----- - --------
x - 3          2
        (x - 3)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify