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Derivative of:
Derivative of x^x^x
Derivative of x^-11
Derivative of (x+5)/(x+1)
Derivative of x^(4/3)
Identical expressions
a*(cos two y)^(one /2)
a multiply by ( co sinus of e of 2y) to the power of (1 divide by 2)
a multiply by ( co sinus of e of two y) to the power of (one divide by 2)
a*(cos2y)(1/2)
a*cos2y1/2
a(cos2y)^(1/2)
a(cos2y)(1/2)
acos2y1/2
acos2y^1/2
a*(cos2y)^(1 divide by 2)

The derivative of the function/ a*(cos2y)^(1/2)

Derivative of a*(cos2y)^(1/2)

The solution

You have entered [src]

    __________
a*\/ cos(2*y)

$$a \sqrt{\cos{\left(2 y \right)}}$$

d /    __________\
--\a*\/ cos(2*y) /
dy

$$\frac{\partial}{\partial y} a \sqrt{\cos{\left(2 y \right)}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos{\left(2 y \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \cos{\left(2 y \right)}$ :
  1. Let $u = 2 y$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d y} 2 y$ :
    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: $2$
    The result of the chain rule is:
    
    $- 2 \sin{\left(2 y \right)}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(2 y \right)}}{\sqrt{\cos{\left(2 y \right)}}}$
So, the result is: $- \frac{a \sin{\left(2 y \right)}}{\sqrt{\cos{\left(2 y \right)}}}$

The answer is:

$- \frac{a \sin{\left(2 y \right)}}{\sqrt{\cos{\left(2 y \right)}}}$

The first derivative [src]

-a*sin(2*y) 
------------
  __________
\/ cos(2*y)

$$- \frac{a \sin{\left(2 y \right)}}{\sqrt{\cos{\left(2 y \right)}}}$$

Simplify

The second derivative [src]

   /                     2      \
   |    __________    sin (2*y) |
-a*|2*\/ cos(2*y)  + -----------|
   |                    3/2     |
   \                 cos   (2*y)/

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

Simplify

The third derivative [src]

   /         2     \          
   |    3*sin (2*y)|          
-a*|2 + -----------|*sin(2*y) 
   |        2      |          
   \     cos (2*y) /          
------------------------------
           __________         
         \/ cos(2*y)

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

Simplify

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Derivative of a*(cos2y)^(1/2)

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