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Integral of d{x}:
y/(sqrt(5+y^2))
Identical expressions
y/(sqrt(five +y^ two))
y divide by ( square root of (5 plus y squared ))
y divide by ( square root of (five plus 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

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

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 /

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

Simplify

The graph

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

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The solution