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Identical expressions
sqrt(x)/(two *x+ one)
square root of (x) divide by (2 multiply by x plus 1)
square root of (x) divide by (two multiply by x plus one)
√(x)/(2*x+1)
sqrt(x)/(2x+1)
sqrtx/2x+1
sqrt(x) divide by (2*x+1)
Similar expressions
sqrt(x)/(2*x-1)

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

Derivative of sqrt(x)/(2*x+1)

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                  ___  
    1             2           8*\/ x   
- ------ - --------------- + ----------
     3/2     ___                      2
  4*x      \/ x *(1 + 2*x)   (1 + 2*x) 
---------------------------------------
                1 + 2*x

$$\frac{\frac{8 \sqrt{x}}{\left(2 x + 1\right)^{2}} - \frac{2}{\sqrt{x} \left(2 x + 1\right)} - \frac{1}{4 x^{\frac{3}{2}}}}{2 x + 1}$$

Simplify

The third derivative [src]

  /                                  ___                    \
  |  1             1            16*\/ x            4        |
3*|------ + ---------------- - ---------- + ----------------|
  |   5/2      3/2                      3     ___          2|
  \8*x      2*x   *(1 + 2*x)   (1 + 2*x)    \/ x *(1 + 2*x) /
-------------------------------------------------------------
                           1 + 2*x

$$\frac{3 \left(- \frac{16 \sqrt{x}}{\left(2 x + 1\right)^{3}} + \frac{4}{\sqrt{x} \left(2 x + 1\right)^{2}} + \frac{1}{2 x^{\frac{3}{2}} \left(2 x + 1\right)} + \frac{1}{8 x^{\frac{5}{2}}}\right)}{2 x + 1}$$

Simplify

The graph

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Derivative of sqrt(x)/(2*x+1)

The graph:

Piecewise:

The solution