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Identical expressions
(y×sqrt(sixteen -y^ two))
(y× square root of (16 minus y squared ))
(y× square root of (sixteen minus y to the power of two))
(y×√(16-y^2))
(y×sqrt(16-y2))
y×sqrt16-y2
(y×sqrt(16-y²))
(y×sqrt(16-y to the power of 2))
y×sqrt16-y^2
Similar expressions
(y×sqrt(16+y^2))

The derivative of the function/ (y×sqrt(16-y^2))

Derivative of (y×sqrt(16-y^2))

The solution

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     _________
    /       2 
y*\/  16 - y

$$y \sqrt{16 - y^{2}}$$

y*sqrt(16 - y^2)

Detail solution

Apply the product rule:

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

$f{\left(y \right)} = y$ ; to find $\frac{d}{d y} f{\left(y \right)}$ :
1. Apply the power rule: $y$ goes to $1$
$g{\left(y \right)} = \sqrt{16 - y^{2}}$ ; to find $\frac{d}{d y} g{\left(y \right)}$ :
1. Let $u = 16 - y^{2}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \left(16 - y^{2}\right)$ :
  1. Differentiate $16 - y^{2}$ term by term:
    1. The derivative of the constant $16$ 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: $y^{2}$ goes to $2 y$
      So, the result is: $- 2 y$
    The result is: $- 2 y$
  The result of the chain rule is:
  
  $- \frac{y}{\sqrt{16 - y^{2}}}$
The result is: $- \frac{y^{2}}{\sqrt{16 - y^{2}}} + \sqrt{16 - y^{2}}$
Now simplify:

$\frac{2 \left(8 - y^{2}\right)}{\sqrt{16 - y^{2}}}$

The answer is:

$\frac{2 \left(8 - y^{2}\right)}{\sqrt{16 - y^{2}}}$

The graph

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The first derivative [src]

   _________         2     
  /       2         y      
\/  16 - y   - ------------
                  _________
                 /       2 
               \/  16 - y

$$- \frac{y^{2}}{\sqrt{16 - y^{2}}} + \sqrt{16 - y^{2}}$$

Simplify

The second derivative [src]

  /         2   \
  |        y    |
y*|-3 + --------|
  |            2|
  \     -16 + y /
-----------------
      _________  
     /       2   
   \/  16 - y

$$\frac{y \left(\frac{y^{2}}{y^{2} - 16} - 3\right)}{\sqrt{16 - y^{2}}}$$

Simplify

The third derivative [src]

  /        2  \ /         2   \
  |       y   | |        y    |
3*|1 + -------|*|-1 + --------|
  |          2| |            2|
  \    16 - y / \     -16 + y /
-------------------------------
             _________         
            /       2          
          \/  16 - y

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

Simplify

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Derivative of (y×sqrt(16-y^2))

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The solution