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Derivative of:
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Graphing y =:
(2*x+3)/(x-1)
Identical expressions
(two *x+ three)/(x- one)
(2 multiply by x plus 3) divide by (x minus 1)
(two multiply by x plus three) divide by (x minus one)
(2x+3)/(x-1)
2x+3/x-1
(2*x+3) divide by (x-1)
Similar expressions
(2*x-3)/(x-1)
(2*x+3)/(x+1)

The derivative of the function/ (2*x+3)/(x-1)

Derivative of (2*x+3)/(x-1)

The solution

You have entered [src]

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

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

d /2*x + 3\
--|-------|
dx\ x - 1 /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of (2*x+3)/(x-1)