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Graphing y =:
1/(sin(2*x)^2)
Identical expressions
one /(sin(two *x)^ two)
1 divide by ( sinus of (2 multiply by x) squared )
one divide by ( sinus of (two multiply by x) to the power of two)
1/(sin(2*x)2)
1/sin2*x2
1/(sin(2*x)²)
1/(sin(2*x) to the power of 2)
1/(sin(2x)^2)
1/(sin(2x)2)
1/sin2x2
1/sin2x^2
1 divide by (sin(2*x)^2)

The derivative of the function/ 1/(sin(2*x)^2)

Derivative of 1/(sin(2*x)^2)

The solution

You have entered [src]

    1    
---------
   2     
sin (2*x)

$$\frac{1}{\sin^{2}{\left(2 x \right)}}$$

1/(sin(2*x)^2)

Detail solution

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

   -4*cos(2*x)    
------------------
            2     
sin(2*x)*sin (2*x)

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

Simplify

The second derivative [src]

  /         2     \
  |    3*cos (2*x)|
8*|1 + -----------|
  |        2      |
  \     sin (2*x) /
-------------------
        2          
     sin (2*x)

$$\frac{8 \left(1 + \frac{3 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right)}{\sin^{2}{\left(2 x \right)}}$$

Simplify

The third derivative [src]

    /         2     \         
    |    3*cos (2*x)|         
-64*|2 + -----------|*cos(2*x)
    |        2      |         
    \     sin (2*x) /         
------------------------------
             3                
          sin (2*x)

$$- \frac{64 \left(2 + \frac{3 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \cos{\left(2 x \right)}}{\sin^{3}{\left(2 x \right)}}$$

Simplify

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Identical expressions

Derivative of 1/(sin(2*x)^2)

The graph:

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The solution