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Derivative of 1/(2*sqrt(x))
Identical expressions
(two x- one)/(3x^2+ four)
(2x minus 1) divide by (3x squared plus 4)
(two x minus one) divide by (3x squared plus four)
(2x-1)/(3x2+4)
2x-1/3x2+4
(2x-1)/(3x²+4)
(2x-1)/(3x to the power of 2+4)
2x-1/3x^2+4
(2x-1) divide by (3x^2+4)
Similar expressions
(2x+1)/(3x^2+4)
(2x-1)/(3x^2-4)

The derivative of the function/ (2x-1)/(3x^2+4)

Derivative of (2x-1)/(3x^2+4)

The solution

You have entered [src]

2*x - 1 
--------
   2    
3*x  + 4

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

(2*x - 1)/(3*x^2 + 4)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x - 1$ and $g{\left(x \right)} = 3 x^{2} + 4$ .

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 $3 x^{2} + 4$ term by term:
  1. The derivative of the constant $4$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $6 x$
  The result is: $6 x$
Now plug in to the quotient rule:

$\frac{6 x^{2} - 6 x \left(2 x - 1\right) + 8}{\left(3 x^{2} + 4\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2       6*x*(2*x - 1)
-------- - -------------
   2                  2 
3*x  + 4    /   2    \  
            \3*x  + 4/

$$- \frac{6 x \left(2 x - 1\right)}{\left(3 x^{2} + 4\right)^{2}} + \frac{2}{3 x^{2} + 4}$$

Simplify

The second derivative [src]

  /                  /          2  \\
  |                  |      12*x   ||
6*|-4*x + (-1 + 2*x)*|-1 + --------||
  |                  |            2||
  \                  \     4 + 3*x //
-------------------------------------
                       2             
             /       2\              
             \4 + 3*x /

$$\frac{6 \left(- 4 x + \left(2 x - 1\right) \left(\frac{12 x^{2}}{3 x^{2} + 4} - 1\right)\right)}{\left(3 x^{2} + 4\right)^{2}}$$

Simplify

The third derivative [src]

   /                               /          2  \\
   |                               |       6*x   ||
   |                6*x*(-1 + 2*x)*|-1 + --------||
   |          2                    |            2||
   |      12*x                     \     4 + 3*x /|
36*|-1 + -------- - ------------------------------|
   |            2                     2           |
   \     4 + 3*x               4 + 3*x            /
---------------------------------------------------
                              2                    
                    /       2\                     
                    \4 + 3*x /

$$\frac{36 \left(\frac{12 x^{2}}{3 x^{2} + 4} - \frac{6 x \left(2 x - 1\right) \left(\frac{6 x^{2}}{3 x^{2} + 4} - 1\right)}{3 x^{2} + 4} - 1\right)}{\left(3 x^{2} + 4\right)^{2}}$$

Simplify

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Derivative of (2x-1)/(3x^2+4)

The graph:

Piecewise:

The solution