Derivative of 1/sqrtsin5x

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     1      
------------
  __________
\/ sin(5*x)

$$\frac{1}{\sqrt{\sin{\left(5 x \right)}}}$$

1/(sqrt(sin(5*x)))

Detail solution

Let $u = \sqrt{\sin{\left(5 x \right)}}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\sin{\left(5 x \right)}}$ :
1. Let $u = \sin{\left(5 x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    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: $5$
    The result of the chain rule is:
    
    $5 \cos{\left(5 x \right)}$
  The result of the chain rule is:
  
  $\frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$
The result of the chain rule is:

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

The answer is:

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

The graph

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The first derivative [src]

      -5*cos(5*x)      
-----------------------
             __________
2*sin(5*x)*\/ sin(5*x)

$$- \frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}} \sin{\left(5 x \right)}}$$

Simplify

The second derivative [src]

   /         2     \
   |    3*cos (5*x)|
25*|2 + -----------|
   |        2      |
   \     sin (5*x) /
--------------------
       __________   
   4*\/ sin(5*x)

$$\frac{25 \left(2 + \frac{3 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right)}{4 \sqrt{\sin{\left(5 x \right)}}}$$

Simplify

The third derivative [src]

     /           2     \         
     |     15*cos (5*x)|         
-125*|14 + ------------|*cos(5*x)
     |         2       |         
     \      sin (5*x)  /         
---------------------------------
               3/2               
          8*sin   (5*x)

$$- \frac{125 \left(14 + \frac{15 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{8 \sin^{\frac{3}{2}}{\left(5 x \right)}}$$

Simplify

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Derivative of 1/sqrtsin5x

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The solution