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Derivative of:
Derivative of y=3x²+5x+4
Derivative of sin(x)+3
Derivative of e^cot(x)
Derivative of (x+3)/(x-3)
Identical expressions
sqrt(five)((2x- one)/(2x+ one))
square root of (5)((2x minus 1) divide by (2x plus 1))
square root of (five)((2x minus one) divide by (2x plus one))
√(5)((2x-1)/(2x+1))
sqrt52x-1/2x+1
sqrt(5)((2x-1) divide by (2x+1))
Similar expressions
sqrt(5)((2x-1)/(2x-1))
sqrt(5)((2x+1)/(2x+1))

The derivative of the function/ sqrt(5)((2x-1)/(2x+1))

Derivative of sqrt(5)((2x-1)/(2x+1))

The solution

You have entered [src]

  ___ 2*x - 1
\/ 5 *-------
      2*x + 1

$$\sqrt{5} \frac{2 x - 1}{2 x + 1}$$

sqrt(5)*((2*x - 1)/(2*x + 1))

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. 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:
  
  $\frac{4}{\left(2 x + 1\right)^{2}}$
So, the result is: $\frac{4 \sqrt{5}}{\left(2 x + 1\right)^{2}}$

The answer is:

$\frac{4 \sqrt{5}}{\left(2 x + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

    ___ /     -1 + 2*x\
8*\/ 5 *|-1 + --------|
        \     1 + 2*x /
-----------------------
                2      
       (1 + 2*x)

$$\frac{8 \sqrt{5} \left(\frac{2 x - 1}{2 x + 1} - 1\right)}{\left(2 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

      ___ /     -1 + 2*x\
-48*\/ 5 *|-1 + --------|
          \     1 + 2*x /
-------------------------
                 3       
        (1 + 2*x)

$$- \frac{48 \sqrt{5} \left(\frac{2 x - 1}{2 x + 1} - 1\right)}{\left(2 x + 1\right)^{3}}$$

Simplify