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Integral of d{x}:
x(4-x^2)^(1/2)
Identical expressions
x(four -x^ two)^(one / two)
x(4 minus x squared ) to the power of (1 divide by 2)
x(four minus x to the power of two) to the power of (one divide by two)
x(4-x2)(1/2)
x4-x21/2
x(4-x²)^(1/2)
x(4-x to the power of 2) to the power of (1/2)
x4-x^2^1/2
x(4-x^2)^(1 divide by 2)
Similar expressions
x(4+x^2)^(1/2)

The derivative of the function/ x(4-x^2)^(1/2)

Derivative of x(4-x^2)^(1/2)

The solution

You have entered [src]

     ________
    /      2 
x*\/  4 - x

$$x \sqrt{4 - x^{2}}$$

x*sqrt(4 - x^2)

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

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

The answer is:

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

The graph

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The first derivative [src]

   ________         2    
  /      2         x     
\/  4 - x   - -----------
                 ________
                /      2 
              \/  4 - x

$$- \frac{x^{2}}{\sqrt{4 - x^{2}}} + \sqrt{4 - x^{2}}$$

Simplify

The second derivative [src]

  /         2  \
  |        x   |
x*|-3 + -------|
  |           2|
  \     -4 + x /
----------------
     ________   
    /      2    
  \/  4 - x

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

Simplify

The third derivative [src]

  /       2  \ /         2  \
  |      x   | |        x   |
3*|1 + ------|*|-1 + -------|
  |         2| |           2|
  \    4 - x / \     -4 + x /
-----------------------------
            ________         
           /      2          
         \/  4 - x

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

Simplify

The graph

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Derivative of x(4-x^2)^(1/2)

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The solution