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

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

The solution

You have entered [src]

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

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

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

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 $g{\left(x \right)} = 2 x + 1$ .

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 $2 x + 1$ 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: $2$
Now plug in to the quotient rule:

$\frac{4}{\left(2 x + 1\right)^{2}}$

The answer is:

$\frac{4}{\left(2 x + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /     -1 + 2*x\
8*|-1 + --------|
  \     1 + 2*x /
-----------------
             2   
    (1 + 2*x)

$$\frac{8 \left(\frac{2 x - 1}{2 x + 1} - 1\right)}{\left(2 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

   /    -1 + 2*x\
48*|1 - --------|
   \    1 + 2*x /
-----------------
             3   
    (1 + 2*x)

$$\frac{48 \left(- \frac{2 x - 1}{2 x + 1} + 1\right)}{\left(2 x + 1\right)^{3}}$$

Simplify

The graph

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