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Identical expressions
y/(sqrt(five -y^ two))
y divide by ( square root of (5 minus y squared ))
y divide by ( square root of (five minus y to the power of two))
y/(√(5-y^2))
y/(sqrt(5-y2))
y/sqrt5-y2
y/(sqrt(5-y²))
y/(sqrt(5-y to the power of 2))
y/sqrt5-y^2
y divide by (sqrt(5-y^2))
Similar expressions
y/(sqrt(5+y^2))

The derivative of the function/ y/(sqrt(5-y^2))

Derivative of y/(sqrt(5-y^2))

The solution

You have entered [src]

     y     
-----------
   ________
  /      2 
\/  5 - y

$$\frac{y}{\sqrt{5 - y^{2}}}$$

y/sqrt(5 - y^2)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d y} \frac{f{\left(y \right)}}{g{\left(y \right)}} = \frac{- f{\left(y \right)} \frac{d}{d y} g{\left(y \right)} + g{\left(y \right)} \frac{d}{d y} f{\left(y \right)}}{g^{2}{\left(y \right)}}$

$f{\left(y \right)} = y$ and $g{\left(y \right)} = \sqrt{5 - y^{2}}$ .

To find $\frac{d}{d y} f{\left(y \right)}$ :
1. Apply the power rule: $y$ goes to $1$
To find $\frac{d}{d y} g{\left(y \right)}$ :
1. Let $u = 5 - y^{2}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \left(5 - y^{2}\right)$ :
  1. Differentiate $5 - y^{2}$ term by term:
    1. The derivative of the constant $5$ 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:
  
  $- \frac{y}{\sqrt{5 - y^{2}}}$
Now plug in to the quotient rule:

$\frac{\frac{y^{2}}{\sqrt{5 - y^{2}}} + \sqrt{5 - y^{2}}}{5 - y^{2}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

                    2    
     1             y     
----------- + -----------
   ________           3/2
  /      2    /     2\   
\/  5 - y     \5 - y /

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

Simplify

The second derivative [src]

  /         2 \
  |      3*y  |
y*|3 - -------|
  |          2|
  \    -5 + y /
---------------
          3/2  
  /     2\     
  \5 - y /

$$\frac{y \left(- \frac{3 y^{2}}{y^{2} - 5} + 3\right)}{\left(5 - y^{2}\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

   /                  /          2 \\
   |                2 |       5*y  ||
   |               y *|-3 + -------||
   |          2       |           2||
   |       3*y        \     -5 + y /|
-3*|-1 + ------- + -----------------|
   |           2              2     |
   \     -5 + y          5 - y      /
-------------------------------------
                     3/2             
             /     2\                
             \5 - y /

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

Simplify

The graph

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Derivative of y/(sqrt(5-y^2))

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The solution