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Identical expressions
y=(x-(3x- one)^ one / two)^ one / two
y equally (x minus (3x minus 1) to the power of 1 divide by 2) to the power of 1 divide by 2
y equally (x minus (3x minus one) to the power of one divide by two) to the power of one divide by two
y=(x-(3x-1)1/2)1/2
y=x-3x-11/21/2
y=x-3x-1^1/2^1/2
y=(x-(3x-1)^1 divide by 2)^1 divide by 2
Similar expressions
y=(x-(3x+1)^1/2)^1/2
y=(x+(3x-1)^1/2)^1/2

The derivative of the function/ y=(x-(3x-1)^1/2)^1/2

Derivative of y=(x-(3x-1)^1/2)^1/2

The solution

You have entered [src]

   _________________
  /       _________ 
\/  x - \/ 3*x - 1

$$\sqrt{x - \sqrt{3 x - 1}}$$

sqrt(x - sqrt(3*x - 1))

Detail solution

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

$\frac{1 - \frac{3}{2 \sqrt{3 x - 1}}}{2 \sqrt{x - \sqrt{3 x - 1}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                  2
                /         3      \ 
                |2 - ------------| 
                |      __________| 
      18        \    \/ -1 + 3*x / 
------------- - -------------------
          3/2           __________ 
(-1 + 3*x)        x - \/ -1 + 3*x  
-----------------------------------
            __________________     
           /       __________      
      16*\/  x - \/ -1 + 3*x

$$\frac{- \frac{\left(2 - \frac{3}{\sqrt{3 x - 1}}\right)^{2}}{x - \sqrt{3 x - 1}} + \frac{18}{\left(3 x - 1\right)^{\frac{3}{2}}}}{16 \sqrt{x - \sqrt{3 x - 1}}}$$

Simplify

The third derivative [src]

  /                                    3                                   \
  |                  /         3      \            /         3      \      |
  |                  |2 - ------------|         18*|2 - ------------|      |
  |                  |      __________|            |      __________|      |
  |       108        \    \/ -1 + 3*x /            \    \/ -1 + 3*x /      |
3*|- ------------- + ------------------- - --------------------------------|
  |            5/2                     2             3/2 /      __________\|
  |  (-1 + 3*x)      /      __________\    (-1 + 3*x)   *\x - \/ -1 + 3*x /|
  \                  \x - \/ -1 + 3*x /                                    /
----------------------------------------------------------------------------
                                __________________                          
                               /       __________                           
                          64*\/  x - \/ -1 + 3*x

$$\frac{3 \left(\frac{\left(2 - \frac{3}{\sqrt{3 x - 1}}\right)^{3}}{\left(x - \sqrt{3 x - 1}\right)^{2}} - \frac{18 \left(2 - \frac{3}{\sqrt{3 x - 1}}\right)}{\left(x - \sqrt{3 x - 1}\right) \left(3 x - 1\right)^{\frac{3}{2}}} - \frac{108}{\left(3 x - 1\right)^{\frac{5}{2}}}\right)}{64 \sqrt{x - \sqrt{3 x - 1}}}$$

Simplify

The graph

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Derivative of y=(x-(3x-1)^1/2)^1/2

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The solution