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sqrt(two *x+ one)/x
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square root of (two multiply by x plus one) divide by x
√(2*x+1)/x
sqrt(2x+1)/x
sqrt2x+1/x
sqrt(2*x+1) divide by x
Similar expressions
y=sqrt3x+sqrt^2x+1/x
sqrt(2*x-1)/x
sqrt3x+sqrt^2x+1/x

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

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

The solution

You have entered [src]

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

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

sqrt(2*x + 1)/x

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 1\right)$ :
  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: $2$
  The result of the chain rule is:
  
  $\frac{1}{\sqrt{2 x + 1}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$\frac{\frac{x}{\sqrt{2 x + 1}} - \sqrt{2 x + 1}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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The solution