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sin^ two (cos3x)+2pi
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sin2(cos3x)+2pi
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Similar expressions
sin^2(cos3x)-2pi

The derivative of the function/ sin^2(cos3x)+2pi

Derivative of sin^2(cos3x)+2pi

The solution

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   2                 
sin (cos(3*x)) + 2*pi

$$\sin^{2}{\left(\cos{\left(3 x \right)} \right)} + 2 \pi$$

sin(cos(3*x))^2 + 2*pi

Detail solution

Differentiate $\sin^{2}{\left(\cos{\left(3 x \right)} \right)} + 2 \pi$ term by term:
1. Let $u = \sin{\left(\cos{\left(3 x \right)} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\cos{\left(3 x \right)} \right)}$ :
  1. Let $u = \cos{\left(3 x \right)}$ .
  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 $\frac{d}{d x} \cos{\left(3 x \right)}$ :
    1. Let $u = 3 x$ .
    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 $\frac{d}{d x} 3 x$ :
      1. 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: $3$
      The result of the chain rule is:
      
      $- 3 \sin{\left(3 x \right)}$
    The result of the chain rule is:
    
    $- 3 \sin{\left(3 x \right)} \cos{\left(\cos{\left(3 x \right)} \right)}$
  The result of the chain rule is:
  
  $- 6 \sin{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)}$
4. The derivative of the constant $2 \pi$ is zero.
The result is: $- 6 \sin{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-6*cos(cos(3*x))*sin(3*x)*sin(cos(3*x))

$$- 6 \sin{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)}$$

Simplify

The second derivative [src]

   /   2              2           2         2                                                 \
18*\cos (cos(3*x))*sin (3*x) - sin (3*x)*sin (cos(3*x)) - cos(3*x)*cos(cos(3*x))*sin(cos(3*x))/

$$18 \left(- \sin^{2}{\left(3 x \right)} \sin^{2}{\left(\cos{\left(3 x \right)} \right)} + \sin^{2}{\left(3 x \right)} \cos^{2}{\left(\cos{\left(3 x \right)} \right)} - \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(3 x \right)} \cos{\left(\cos{\left(3 x \right)} \right)}\right)$$

Simplify

The third derivative [src]

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

$$54 \left(4 \sin^{2}{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)} - 3 \sin^{2}{\left(\cos{\left(3 x \right)} \right)} \cos{\left(3 x \right)} + \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)} + 3 \cos{\left(3 x \right)} \cos^{2}{\left(\cos{\left(3 x \right)} \right)}\right) \sin{\left(3 x \right)}$$

Simplify

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Derivative of sin^2(cos3x)+2pi

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The solution