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Derivative of:
Derivative of log(x)
Derivative of x+1
Derivative of x-2
Derivative of e
Identical expressions
cos(3x)^ six *sin(6x)^ two
co sinus of e of (3x) to the power of 6 multiply by sinus of (6x) squared
co sinus of e of (3x) to the power of six multiply by sinus of (6x) to the power of two
cos(3x)6*sin(6x)2
cos3x6*sin6x2
cos(3x)⁶*sin(6x)²
cos(3x) to the power of 6*sin(6x) to the power of 2
cos(3x)^6sin(6x)^2
cos(3x)6sin(6x)2
cos3x6sin6x2
cos3x^6sin6x^2

The derivative of the function/ cos(3x)^6*sin(6x)^2

Derivative of cos(3x)^6*sin(6x)^2

The solution

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   6         2     
cos (3*x)*sin (6*x)

$$\sin^{2}{\left(6 x \right)} \cos^{6}{\left(3 x \right)}$$

d /   6         2     \
--\cos (3*x)*sin (6*x)/
dx

$$\frac{d}{d x} \sin^{2}{\left(6 x \right)} \cos^{6}{\left(3 x \right)}$$

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \cos^{6}{\left(3 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \cos{\left(3 x \right)}$ .
2. Apply the power rule: $u^{6}$ goes to $6 u^{5}$
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.
      1. 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:
  
  $- 18 \sin{\left(3 x \right)} \cos^{5}{\left(3 x \right)}$
$g{\left(x \right)} = \sin^{2}{\left(6 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(6 x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(6 x \right)}$ :
  1. Let $u = 6 x$ .
  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 $\frac{d}{d x} 6 x$ :
    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: $6$
    The result of the chain rule is:
    
    $6 \cos{\left(6 x \right)}$
  The result of the chain rule is:
  
  $12 \sin{\left(6 x \right)} \cos{\left(6 x \right)}$
The result is: $- 18 \sin{\left(3 x \right)} \sin^{2}{\left(6 x \right)} \cos^{5}{\left(3 x \right)} + 12 \sin{\left(6 x \right)} \cos^{6}{\left(3 x \right)} \cos{\left(6 x \right)}$
Now simplify:

$\frac{3 \left(- 6 \sin{\left(6 x \right)} + 5 \sin{\left(12 x \right)}\right) \left(\cos{\left(6 x \right)} + 1\right)^{3}}{8}$

The answer is:

$\frac{3 \left(- 6 \sin{\left(6 x \right)} + 5 \sin{\left(12 x \right)}\right) \left(\cos{\left(6 x \right)} + 1\right)^{3}}{8}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        5         2                       6                       
- 18*cos (3*x)*sin (6*x)*sin(3*x) + 12*cos (3*x)*cos(6*x)*sin(6*x)

$$- 18 \sin{\left(3 x \right)} \sin^{2}{\left(6 x \right)} \cos^{5}{\left(3 x \right)} + 12 \sin{\left(6 x \right)} \cos^{6}{\left(3 x \right)} \cos{\left(6 x \right)}$$

Simplify

The second derivative [src]

      4      /       2      /   2           2     \        2      /     2             2     \                                         \
18*cos (3*x)*\- 4*cos (3*x)*\sin (6*x) - cos (6*x)/ + 3*sin (6*x)*\- cos (3*x) + 5*sin (3*x)/ - 24*cos(3*x)*cos(6*x)*sin(3*x)*sin(6*x)/

$$18 \cdot \left(3 \cdot \left(5 \sin^{2}{\left(3 x \right)} - \cos^{2}{\left(3 x \right)}\right) \sin^{2}{\left(6 x \right)} - 4 \left(\sin^{2}{\left(6 x \right)} - \cos^{2}{\left(6 x \right)}\right) \cos^{2}{\left(3 x \right)} - 24 \sin{\left(3 x \right)} \sin{\left(6 x \right)} \cos{\left(3 x \right)} \cos{\left(6 x \right)}\right) \cos^{4}{\left(3 x \right)}$$

Simplify

The third derivative [src]

       3      /       3                               2      /       2             2     \                  2      /   2           2     \              /     2             2     \                           \
216*cos (3*x)*\- 8*cos (3*x)*cos(6*x)*sin(6*x) - 3*sin (6*x)*\- 4*cos (3*x) + 5*sin (3*x)/*sin(3*x) + 18*cos (3*x)*\sin (6*x) - cos (6*x)/*sin(3*x) + 9*\- cos (3*x) + 5*sin (3*x)/*cos(3*x)*cos(6*x)*sin(6*x)/

$$216 \left(- 3 \cdot \left(5 \sin^{2}{\left(3 x \right)} - 4 \cos^{2}{\left(3 x \right)}\right) \sin{\left(3 x \right)} \sin^{2}{\left(6 x \right)} + 9 \cdot \left(5 \sin^{2}{\left(3 x \right)} - \cos^{2}{\left(3 x \right)}\right) \sin{\left(6 x \right)} \cos{\left(3 x \right)} \cos{\left(6 x \right)} + 18 \left(\sin^{2}{\left(6 x \right)} - \cos^{2}{\left(6 x \right)}\right) \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)} - 8 \sin{\left(6 x \right)} \cos^{3}{\left(3 x \right)} \cos{\left(6 x \right)}\right) \cos^{3}{\left(3 x \right)}$$

Simplify

The graph

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Derivative of cos(3x)^6*sin(6x)^2

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The solution