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((two x- one)^ two)*sqrt(one -2*x)
((2x minus 1) squared ) multiply by square root of (1 minus 2 multiply by x)
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Similar expressions
((2x-1)^2)*sqrt(1+2*x)
((2x+1)^2)*sqrt(1-2*x)

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

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

The solution

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \left(2 x - 1\right)^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x - 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 1\right)$ :
  1. Differentiate $2 x - 1$ term by term:
    1. 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: $2$
    2. The derivative of the constant $-1$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $8 x - 4$
$g{\left(x \right)} = \sqrt{1 - 2 x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 - 2 x$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - 2 x\right)$ :
  1. Differentiate $1 - 2 x$ 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.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $-2$
    The result is: $-2$
  The result of the chain rule is:
  
  $- \frac{1}{\sqrt{1 - 2 x}}$
The result is: $\sqrt{1 - 2 x} \left(8 x - 4\right) - \frac{\left(2 x - 1\right)^{2}}{\sqrt{1 - 2 x}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

                                   2
  _________               (2*x - 1) 
\/ 1 - 2*x *(-4 + 8*x) - -----------
                           _________
                         \/ 1 - 2*x

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

Simplify

The second derivative [src]

                          2                
    _________   (-1 + 2*x)     8*(-1 + 2*x)
8*\/ 1 - 2*x  - ------------ - ------------
                         3/2     _________ 
                (1 - 2*x)      \/ 1 - 2*x

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

Simplify

The third derivative [src]

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

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

Simplify

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

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Piecewise:

The solution

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

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The solution

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