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Derivative of √x
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Identical expressions
ctg two -((cos^2(8x))/(16sin(16x)))
ctg2 minus (( co sinus of e of squared (8x)) divide by (16 sinus of (16x)))
ctg two minus (( co sinus of e of squared (8x)) divide by (16 sinus of (16x)))
ctg2-((cos2(8x))/(16sin(16x)))
ctg2-cos28x/16sin16x
ctg2-((cos²(8x))/(16sin(16x)))
ctg2-((cos to the power of 2(8x))/(16sin(16x)))
ctg2-cos^28x/16sin16x
ctg2-((cos^2(8x)) divide by (16sin(16x)))
Similar expressions
ctg2+((cos^2(8x))/(16sin(16x)))

The derivative of the function/ ctg2-((cos^2(8x))/(16sin(16x)))

Derivative of ctg2-((cos^2(8x))/(16sin(16x)))

The solution

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             2       
          cos (8*x)  
cot(2) - ------------
         16*sin(16*x)

$$\cot{\left(2 \right)} - \frac{\cos^{2}{\left(8 x \right)}}{16 \sin{\left(16 x \right)}}$$

cot(2) - cos(8*x)^2/(16*sin(16*x))

Detail solution

Differentiate $\cot{\left(2 \right)} - \frac{\cos^{2}{\left(8 x \right)}}{16 \sin{\left(16 x \right)}}$ term by term:
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(2 \right)} = \frac{1}{\tan{\left(2 \right)}}$
  2. The derivative of the constant $\frac{1}{\tan{\left(2 \right)}}$ is zero.
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(2 \right)} = \frac{\cos{\left(2 \right)}}{\sin{\left(2 \right)}}$
  2. The derivative of the constant $\frac{\cos{\left(2 \right)}}{\sin{\left(2 \right)}}$ is zero.
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos^{2}{\left(8 x \right)}$ and $g{\left(x \right)} = 16 \sin{\left(16 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \cos{\left(8 x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(8 x \right)}$ :
      1. Let $u = 8 x$ .
      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 $\frac{d}{d x} 8 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $8$
        
        The result of the chain rule is:
        
        $- 8 \sin{\left(8 x \right)}$
      The result of the chain rule is:
      
      $- 16 \sin{\left(8 x \right)} \cos{\left(8 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Let $u = 16 x$ .
      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 $\frac{d}{d x} 16 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $16$
        
        The result of the chain rule is:
        
        $16 \cos{\left(16 x \right)}$
      So, the result is: $256 \cos{\left(16 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{- 256 \sin{\left(8 x \right)} \sin{\left(16 x \right)} \cos{\left(8 x \right)} - 256 \cos^{2}{\left(8 x \right)} \cos{\left(16 x \right)}}{256 \sin^{2}{\left(16 x \right)}}$
  So, the result is: $- \frac{- 256 \sin{\left(8 x \right)} \sin{\left(16 x \right)} \cos{\left(8 x \right)} - 256 \cos^{2}{\left(8 x \right)} \cos{\left(16 x \right)}}{256 \sin^{2}{\left(16 x \right)}}$
The result is: $- \frac{- 256 \sin{\left(8 x \right)} \sin{\left(16 x \right)} \cos{\left(8 x \right)} - 256 \cos^{2}{\left(8 x \right)} \cos{\left(16 x \right)}}{256 \sin^{2}{\left(16 x \right)}}$
Now simplify:

$\frac{1}{256 \left(2 \sin^{2}{\left(x \right)} - 1\right)^{2} \left(8 \sin^{4}{\left(x \right)} - 8 \sin^{2}{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{1}{256 \left(2 \sin^{2}{\left(x \right)} - 1\right)^{2} \left(8 \sin^{4}{\left(x \right)} - 8 \sin^{2}{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}$

The first derivative [src]

   2                                                   
cos (8*x)*cos(16*x)           1                        
------------------- + 16*------------*cos(8*x)*sin(8*x)
        2                16*sin(16*x)                  
     sin (16*x)

$$16 \sin{\left(8 x \right)} \frac{1}{16 \sin{\left(16 x \right)}} \cos{\left(8 x \right)} + \frac{\cos^{2}{\left(8 x \right)} \cos{\left(16 x \right)}}{\sin^{2}{\left(16 x \right)}}$$

Simplify

The second derivative [src]

   /                             2         2                                      \
   |   2           2        4*cos (8*x)*cos (16*x)   4*cos(8*x)*cos(16*x)*sin(8*x)|
-8*|cos (8*x) + sin (8*x) + ---------------------- + -----------------------------|
   |                                 2                         sin(16*x)          |
   \                              sin (16*x)                                      /
-----------------------------------------------------------------------------------
                                     sin(16*x)

$$- \frac{8 \left(\sin^{2}{\left(8 x \right)} + \frac{4 \sin{\left(8 x \right)} \cos{\left(8 x \right)} \cos{\left(16 x \right)}}{\sin{\left(16 x \right)}} + \cos^{2}{\left(8 x \right)} + \frac{4 \cos^{2}{\left(8 x \right)} \cos^{2}{\left(16 x \right)}}{\sin^{2}{\left(16 x \right)}}\right)}{\sin{\left(16 x \right)}}$$

Simplify

The third derivative [src]

    /                           2                       2                        2         3               2                        \
    |                      3*sin (8*x)*cos(16*x)   7*cos (8*x)*cos(16*x)   12*cos (8*x)*cos (16*x)   12*cos (16*x)*cos(8*x)*sin(8*x)|
128*|4*cos(8*x)*sin(8*x) + --------------------- + --------------------- + ----------------------- + -------------------------------|
    |                            sin(16*x)               sin(16*x)                   3                             2                |
    \                                                                             sin (16*x)                    sin (16*x)          /
-------------------------------------------------------------------------------------------------------------------------------------
                                                              sin(16*x)

$$\frac{128 \left(\frac{3 \sin^{2}{\left(8 x \right)} \cos{\left(16 x \right)}}{\sin{\left(16 x \right)}} + 4 \sin{\left(8 x \right)} \cos{\left(8 x \right)} + \frac{12 \sin{\left(8 x \right)} \cos{\left(8 x \right)} \cos^{2}{\left(16 x \right)}}{\sin^{2}{\left(16 x \right)}} + \frac{7 \cos^{2}{\left(8 x \right)} \cos{\left(16 x \right)}}{\sin{\left(16 x \right)}} + \frac{12 \cos^{2}{\left(8 x \right)} \cos^{3}{\left(16 x \right)}}{\sin^{3}{\left(16 x \right)}}\right)}{\sin{\left(16 x \right)}}$$

Simplify

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Derivative of ctg2-((cos^2(8x))/(16sin(16x)))

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The solution

Method #1

Method #2