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Identical expressions
two *sqrt((one -x)*x)
2 multiply by square root of ((1 minus x) multiply by x)
two multiply by square root of ((one minus x) multiply by x)
2*√((1-x)*x)
2sqrt((1-x)x)
2sqrt1-xx
Similar expressions
2*sqrt((1+x)*x)

The derivative of the function/ 2*sqrt((1-x)*x)

Derivative of 2sqrt((1-x)x)

The solution

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    ___________
2*\/ (1 - x)*x

$$2 \sqrt{x \left(1 - x\right)}$$

2*sqrt((1 - x)*x)

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

    ___________          
2*\/ x*(1 - x) *(1/2 - x)
-------------------------
        x*(1 - x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                   /                                                              3                 2               2\
     _____________ |2     2      -1 + 2*x    -1 + 2*x   5*(-1 + 2*x)    (-1 + 2*x)      3*(-1 + 2*x)    3*(-1 + 2*x) |
-2*\/ -x*(-1 + x) *|- + ------ - -------- - --------- - ------------ - -------------- + ------------- + -------------|
                   |x   -1 + x       2              2   2*x*(-1 + x)      2         2               2      2         |
                   \                x       (-1 + x)                   8*x *(-1 + x)    4*x*(-1 + x)    4*x *(-1 + x)/
----------------------------------------------------------------------------------------------------------------------
                                                      x*(-1 + x)

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

Simplify

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Derivative of 2sqrt((1-x)x)

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The solution

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Derivative of 2*sqrt((1-x)*x)

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Derivative of 2sqrt((1-x)x)