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Identical expressions
y= three /(two sqrt3x-2)
y equally 3 divide by (2 square root of 3x minus 2)
y equally three divide by (two square root of 3x minus 2)
y=3/(2√3x-2)
y=3/2sqrt3x-2
y=3 divide by (2sqrt3x-2)
Similar expressions
y=3/(2sqrt3x+2)

The derivative of the function/ y=3/(2sqrt3x-2)

Derivative of y=3/(2sqrt3x-2)

The solution

You have entered [src]

      3      
-------------
    _____    
2*\/ 3*x  - 2

$$\frac{3}{2 \sqrt{3 x} - 2}$$

3/(2*sqrt(3*x) - 2)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = 2 \sqrt{3 x} - 2$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 \sqrt{3 x} - 2\right)$ :
  1. Differentiate $2 \sqrt{3 x} - 2$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Let $u = 3 x$ .
      2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
        
        The result of the chain rule is:
        
        $\frac{\sqrt{3}}{2 \sqrt{x}}$
      So, the result is: $\frac{\sqrt{3}}{\sqrt{x}}$
    2. The derivative of the constant $-2$ is zero.
    The result is: $\frac{\sqrt{3}}{\sqrt{x}}$
  The result of the chain rule is:
  
  $- \frac{\sqrt{3}}{\sqrt{x} \left(2 \sqrt{3 x} - 2\right)^{2}}$
So, the result is: $- \frac{3 \sqrt{3}}{\sqrt{x} \left(2 \sqrt{3 x} - 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

            ___       
       -3*\/ 3        
----------------------
                     2
  ___ /    _____    \ 
\/ x *\2*\/ 3*x  - 2/

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

Simplify

The second derivative [src]

  /  ___                       \
  |\/ 3             6          |
3*|----- + --------------------|
  |  3/2     /       ___   ___\|
  \ x      x*\-1 + \/ 3 *\/ x //
--------------------------------
                         2      
       /       ___   ___\       
     8*\-1 + \/ 3 *\/ x /

$$\frac{3 \left(\frac{6}{x \left(\sqrt{3} \sqrt{x} - 1\right)} + \frac{\sqrt{3}}{x^{\frac{3}{2}}}\right)}{8 \left(\sqrt{3} \sqrt{x} - 1\right)^{2}}$$

Simplify

The third derivative [src]

   /  ___                                       ___         \
   |\/ 3              6                     6*\/ 3          |
-9*|----- + --------------------- + ------------------------|
   |  5/2    2 /       ___   ___\                          2|
   | x      x *\-1 + \/ 3 *\/ x /    3/2 /       ___   ___\ |
   \                                x   *\-1 + \/ 3 *\/ x / /
-------------------------------------------------------------
                                         2                   
                       /       ___   ___\                    
                    16*\-1 + \/ 3 *\/ x /

$$- \frac{9 \left(\frac{6}{x^{2} \left(\sqrt{3} \sqrt{x} - 1\right)} + \frac{6 \sqrt{3}}{x^{\frac{3}{2}} \left(\sqrt{3} \sqrt{x} - 1\right)^{2}} + \frac{\sqrt{3}}{x^{\frac{5}{2}}}\right)}{16 \left(\sqrt{3} \sqrt{x} - 1\right)^{2}}$$

Simplify

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Derivative of y=3/(2sqrt3x-2)

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