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Identical expressions
(sqrt4x+ one)/(sqrt2x+ three)
( square root of 4x plus 1) divide by ( square root of 2x plus 3)
( square root of 4x plus one) divide by ( square root of 2x plus three)
(√4x+1)/(√2x+3)
sqrt4x+1/sqrt2x+3
(sqrt4x+1) divide by (sqrt2x+3)
Similar expressions
(sqrt4x-1)/(sqrt2x+3)
(sqrt4x+1)/(sqrt2x-3)

The derivative of the function/ (sqrt4x+1)/(sqrt2x+3)

Derivative of (sqrt4x+1)/(sqrt2x+3)

The solution

You have entered [src]

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

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

(sqrt(4*x) + 1)/(sqrt(2*x) + 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 \sqrt{x} + 1$ and $g{\left(x \right)} = \sqrt{2} \sqrt{x} + 3$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 \sqrt{x} + 1$ 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: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    So, the result is: $\frac{1}{\sqrt{x}}$
  The result is: $\frac{1}{\sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\sqrt{2} \sqrt{x} + 3$ term by term:
  1. The derivative of the constant $3$ 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: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    So, the result is: $\frac{\sqrt{2}}{2 \sqrt{x}}$
  The result is: $\frac{\sqrt{2}}{2 \sqrt{x}}$
Now plug in to the quotient rule:

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

$\frac{6 - \sqrt{2}}{4 x^{\frac{3}{2}} + 18 \sqrt{x} + 12 \sqrt{2} x}$

The answer is:

$\frac{6 - \sqrt{2}}{4 x^{\frac{3}{2}} + 18 \sqrt{x} + 12 \sqrt{2} x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                         ___ /  _____    \  
         1             \/ 2 *\\/ 4*x  + 1/  
------------------- - ----------------------
  ___ /  _____    \                        2
\/ x *\\/ 2*x  + 3/       ___ /  _____    \ 
                      2*\/ x *\\/ 2*x  + 3/

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

Simplify

The second derivative [src]

                                               /  ___                      \
                                 /        ___\ |\/ 2             4         |
                                 \1 + 2*\/ x /*|----- + -------------------|
                    ___                        |  3/2     /      ___   ___\|
    1             \/ 2                         \ x      x*\3 + \/ 2 *\/ x //
- ------ - ------------------- + -------------------------------------------
     3/2     /      ___   ___\                 /      ___   ___\            
  2*x      x*\3 + \/ 2 *\/ x /               4*\3 + \/ 2 *\/ x /            
----------------------------------------------------------------------------
                                    ___   ___                               
                              3 + \/ 2 *\/ x

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (sqrt4x+1)/(sqrt2x+3)

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The solution