Derivative of y=sqrt(sin4x)

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  __________
\/ sin(4*x)

$$\sqrt{\sin{\left(4 x \right)}}$$

d /  __________\
--\\/ sin(4*x) /
dx

$$\frac{d}{d x} \sqrt{\sin{\left(4 x \right)}}$$

Transform

Detail solution

Let $u = \sin{\left(4 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(4 x \right)}$ :
1. Let $u = 4 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $4$
  The result of the chain rule is:
  
  $4 \cos{\left(4 x \right)}$
The result of the chain rule is:

$\frac{2 \cos{\left(4 x \right)}}{\sqrt{\sin{\left(4 x \right)}}}$

The answer is:

$\frac{2 \cos{\left(4 x \right)}}{\sqrt{\sin{\left(4 x \right)}}}$

The graph

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The first derivative [src]

 2*cos(4*x) 
------------
  __________
\/ sin(4*x)

$$\frac{2 \cos{\left(4 x \right)}}{\sqrt{\sin{\left(4 x \right)}}}$$

Simplify

The second derivative [src]

   /                     2      \
   |    __________    cos (4*x) |
-4*|2*\/ sin(4*x)  + -----------|
   |                    3/2     |
   \                 sin   (4*x)/

$$- 4 \cdot \left(2 \sqrt{\sin{\left(4 x \right)}} + \frac{\cos^{2}{\left(4 x \right)}}{\sin^{\frac{3}{2}}{\left(4 x \right)}}\right)$$

Simplify

The third derivative [src]

  /         2     \         
  |    3*cos (4*x)|         
8*|2 + -----------|*cos(4*x)
  |        2      |         
  \     sin (4*x) /         
----------------------------
          __________        
        \/ sin(4*x)

$$\frac{8 \cdot \left(2 + \frac{3 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \cos{\left(4 x \right)}}{\sqrt{\sin{\left(4 x \right)}}}$$

Simplify

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Derivative of y=sqrt(sin4x)

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