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Derivative of:
Derivative of (-2)/x^3
Derivative of ln(x^2-4x)
Derivative of e^(3/x)
Derivative of (2*x+1)*(x^2+3*x-1)
Identical expressions
x(one -x)^(one / two)
x(1 minus x) to the power of (1 divide by 2)
x(one minus x) to the power of (one divide by two)
x(1-x)(1/2)
x1-x1/2
x1-x^1/2
x(1-x)^(1 divide by 2)
Similar expressions
x(1+x)^(1/2)

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

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

The solution

You have entered [src]

    _______
x*\/ 1 - x

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

x*sqrt(1 - x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  _______        x     
\/ 1 - x  - -----------
                _______
            2*\/ 1 - x

$$- \frac{x}{2 \sqrt{1 - x}} + \sqrt{1 - x}$$

Simplify

The second derivative [src]

 /        x    \ 
-|1 + ---------| 
 \    4*(1 - x)/ 
-----------------
      _______    
    \/ 1 - x

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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