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a*y*sqrt(one -(y/b)^ two)
a multiply by y multiply by square root of (1 minus (y divide by b) squared )
a multiply by y multiply by square root of (one minus (y divide by b) to the power of two)
a*y*√(1-(y/b)^2)
a*y*sqrt(1-(y/b)2)
a*y*sqrt1-y/b2
a*y*sqrt(1-(y/b)²)
a*y*sqrt(1-(y/b) to the power of 2)
aysqrt(1-(y/b)^2)
aysqrt(1-(y/b)2)
aysqrt1-y/b2
aysqrt1-y/b^2
a*y*sqrt(1-(y divide by b)^2)
Similar expressions
a*y*sqrt(1+(y/b)^2)

The derivative of the function/ a*y*sqrt(1-(y/b)^2)

Derivative of aysqrt(1-(y/b)^2)

The solution

You have entered [src]

         __________
        /        2 
       /      /y\  
a*y*  /   1 - |-|  
    \/        \b/

$$a y \sqrt{1 - \left(\frac{y}{b}\right)^{2}}$$

(a*y)*sqrt(1 - (y/b)^2)

Detail solution

Apply the product rule:

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

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

$\frac{a \left(b^{2} - 2 y^{2}\right)}{b^{2} \sqrt{\frac{b^{2} - y^{2}}{b^{2}}}}$

The answer is:

$\frac{a \left(b^{2} - 2 y^{2}\right)}{b^{2} \sqrt{\frac{b^{2} - y^{2}}{b^{2}}}}$

The first derivative [src]

       __________                     
      /        2              2       
     /      /y\            a*y        
a*  /   1 - |-|   - ------------------
  \/        \b/             __________
                           /        2 
                     2    /      /y\  
                    b *  /   1 - |-|  
                       \/        \b/

$$a \sqrt{1 - \left(\frac{y}{b}\right)^{2}} - \frac{a y^{2}}{b^{2} \sqrt{1 - \left(\frac{y}{b}\right)^{2}}}$$

Simplify

The second derivative [src]

     /          2    \ 
     |         y     | 
-a*y*|3 + -----------| 
     |       /     2\| 
     |     2 |    y || 
     |    b *|1 - --|| 
     |       |     2|| 
     \       \    b // 
-----------------------
            ________   
           /      2    
    2     /      y     
   b *   /   1 - --    
        /         2    
      \/         b

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

Simplify

The third derivative [src]

                      2
     /          2    \ 
     |         y     | 
-3*a*|1 + -----------| 
     |       /     2\| 
     |     2 |    y || 
     |    b *|1 - --|| 
     |       |     2|| 
     \       \    b // 
-----------------------
            ________   
           /      2    
    2     /      y     
   b *   /   1 - --    
        /         2    
      \/         b

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

Simplify

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Derivative of aysqrt(1-(y/b)^2)

The graph:

Piecewise:

The solution

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Derivative of a*y*sqrt(1-(y/b)^2)

The graph:

Piecewise:

The solution

Derivative of aysqrt(1-(y/b)^2)