Derivative of sinsqrt4x^-1

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     1      
------------
   /  _____\
sin\\/ 4*x /

$$\frac{1}{\sin{\left(\sqrt{4 x} \right)}}$$

1/sin(sqrt(4*x))

Detail solution

Let $u = \sin{\left(\sqrt{4 x} \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\sqrt{4 x} \right)}$ :
1. Let $u = \sqrt{4 x}$ .
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 $\frac{d}{d x} \sqrt{4 x}$ :
  1. Let $u = 4 x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
    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:
    
    $\frac{1}{\sqrt{x}}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(\sqrt{4 x} \right)}}{\sqrt{x}}$
The result of the chain rule is:

$- \frac{\cos{\left(\sqrt{4 x} \right)}}{\sqrt{x} \sin^{2}{\left(\sqrt{4 x} \right)}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

       /  _____\   
   -cos\\/ 4*x /   
-------------------
  ___    2/  _____\
\/ x *sin \\/ 4*x /

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

Simplify

The second derivative [src]

           /  _____\           2/  _____\
1       cos\\/ 4*x /      2*cos \\/ 4*x /
- + ------------------- + ---------------
x      3/2    /  _____\        2/  _____\
    2*x   *sin\\/ 4*x /   x*sin \\/ 4*x /
-----------------------------------------
                  /  _____\              
               sin\\/ 4*x /

$$\frac{\frac{1}{x} + \frac{2 \cos^{2}{\left(\sqrt{4 x} \right)}}{x \sin^{2}{\left(\sqrt{4 x} \right)}} + \frac{\cos{\left(\sqrt{4 x} \right)}}{2 x^{\frac{3}{2}} \sin{\left(\sqrt{4 x} \right)}}}{\sin{\left(\sqrt{4 x} \right)}}$$

Simplify

The third derivative [src]

 /            2/  _____\           /  _____\          3/  _____\             /  _____\  \ 
 | 3     3*cos \\/ 4*x /      5*cos\\/ 4*x /     6*cos \\/ 4*x /        3*cos\\/ 4*x /  | 
-|---- + ---------------- + ----------------- + ------------------ + -------------------| 
 |   2    2    2/  _____\    3/2    /  _____\    3/2    3/  _____\      5/2    /  _____\| 
 \2*x    x *sin \\/ 4*x /   x   *sin\\/ 4*x /   x   *sin \\/ 4*x /   4*x   *sin\\/ 4*x // 
------------------------------------------------------------------------------------------
                                          /  _____\                                       
                                       sin\\/ 4*x /

$$- \frac{\frac{3}{2 x^{2}} + \frac{3 \cos^{2}{\left(\sqrt{4 x} \right)}}{x^{2} \sin^{2}{\left(\sqrt{4 x} \right)}} + \frac{5 \cos{\left(\sqrt{4 x} \right)}}{x^{\frac{3}{2}} \sin{\left(\sqrt{4 x} \right)}} + \frac{6 \cos^{3}{\left(\sqrt{4 x} \right)}}{x^{\frac{3}{2}} \sin^{3}{\left(\sqrt{4 x} \right)}} + \frac{3 \cos{\left(\sqrt{4 x} \right)}}{4 x^{\frac{5}{2}} \sin{\left(\sqrt{4 x} \right)}}}{\sin{\left(\sqrt{4 x} \right)}}$$

Simplify

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Derivative of sinsqrt4x^-1

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