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(sqrt(four *x+ one))/x^ two
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Similar expressions
(sqrt(4*x-1))/x^2

The derivative of the function/ (sqrt(4*x+1))/x^2

Derivative of (sqrt(4*x+1))/x^2

The solution

You have entered [src]

  _________
\/ 4*x + 1 
-----------
      2    
     x

$$\frac{\sqrt{4 x + 1}}{x^{2}}$$

sqrt(4*x + 1)/x^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt{4 x + 1}$ and $g{\left(x \right)} = x^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 4 x + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 1\right)$ :
  1. Differentiate $4 x + 1$ 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: $4$
    The result is: $4$
  The result of the chain rule is:
  
  $\frac{2}{\sqrt{4 x + 1}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
Now plug in to the quotient rule:

$\frac{\frac{2 x^{2}}{\sqrt{4 x + 1}} - 2 x \sqrt{4 x + 1}}{x^{4}}$
Now simplify:

$- \frac{6 x + 2}{x^{3} \sqrt{4 x + 1}}$

The answer is:

$- \frac{6 x + 2}{x^{3} \sqrt{4 x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      _________                 
  2*\/ 4*x + 1          2       
- ------------- + --------------
         3         2   _________
        x         x *\/ 4*x + 1

$$\frac{2}{x^{2} \sqrt{4 x + 1}} - \frac{2 \sqrt{4 x + 1}}{x^{3}}$$

Simplify

The second derivative [src]

  /                                     _________\
  |       2               4         3*\/ 1 + 4*x |
2*|- ------------ - ------------- + -------------|
  |           3/2       _________          2     |
  \  (1 + 4*x)      x*\/ 1 + 4*x          x      /
--------------------------------------------------
                         2                        
                        x

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

Simplify

The third derivative [src]

   /                   _________                                  \
   |     2         2*\/ 1 + 4*x          2                3       |
12*|------------ - ------------- + -------------- + --------------|
   |         5/2          3                   3/2    2   _________|
   \(1 + 4*x)            x         x*(1 + 4*x)      x *\/ 1 + 4*x /
-------------------------------------------------------------------
                                  2                                
                                 x

$$\frac{12 \left(\frac{2}{\left(4 x + 1\right)^{\frac{5}{2}}} + \frac{2}{x \left(4 x + 1\right)^{\frac{3}{2}}} + \frac{3}{x^{2} \sqrt{4 x + 1}} - \frac{2 \sqrt{4 x + 1}}{x^{3}}\right)}{x^{2}}$$

Simplify

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Derivative of (sqrt(4*x+1))/x^2

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The solution