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Derivative of 3xcos(x/3)-9sin(x/3)
Derivative of tg²x
Derivative of 6x^3
Derivative of shx
Identical expressions
three xcos(x/ three)-9sin(x/3)
3x co sinus of e of (x divide by 3) minus 9 sinus of (x divide by 3)
three x co sinus of e of (x divide by three) minus 9 sinus of (x divide by 3)
3xcosx/3-9sinx/3
3xcos(x divide by 3)-9sin(x divide by 3)
Similar expressions
3xcos(x/3)+9sin(x/3)

The derivative of the function/ 3xcos(x/3)-9sin(x/3)

Derivative of 3xcos(x/3)-9sin(x/3)

The solution

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       /x\        /x\
3*x*cos|-| - 9*sin|-|
       \3/        \3/

$$3 x \cos{\left(\frac{x}{3} \right)} - 9 \sin{\left(\frac{x}{3} \right)}$$

d /       /x\        /x\\
--|3*x*cos|-| - 9*sin|-||
dx\       \3/        \3//

$$\frac{d}{d x} \left(3 x \cos{\left(\frac{x}{3} \right)} - 9 \sin{\left(\frac{x}{3} \right)}\right)$$

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Detail solution

Differentiate $3 x \cos{\left(\frac{x}{3} \right)} - 9 \sin{\left(\frac{x}{3} \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = \cos{\left(\frac{x}{3} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \frac{x}{3}$ .
    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 $\frac{d}{d x} \frac{x}{3}$ :
      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: $\frac{1}{3}$
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\frac{x}{3} \right)}}{3}$
    The result is: $- \frac{x \sin{\left(\frac{x}{3} \right)}}{3} + \cos{\left(\frac{x}{3} \right)}$
  So, the result is: $- x \sin{\left(\frac{x}{3} \right)} + 3 \cos{\left(\frac{x}{3} \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \frac{x}{3}$ .
    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 $\frac{d}{d x} \frac{x}{3}$ :
      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: $\frac{1}{3}$
      The result of the chain rule is:
      
      $\frac{\cos{\left(\frac{x}{3} \right)}}{3}$
    So, the result is: $3 \cos{\left(\frac{x}{3} \right)}$
  So, the result is: $- 3 \cos{\left(\frac{x}{3} \right)}$
The result is: $- x \sin{\left(\frac{x}{3} \right)}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

      /x\
-x*sin|-|
      \3/

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

Simplify

The second derivative [src]

 /     /x\         \
 |x*cos|-|         |
 |     \3/      /x\|
-|-------- + sin|-||
 \   3          \3//

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

Simplify

The third derivative [src]

       /x\        /x\
- 6*cos|-| + x*sin|-|
       \3/        \3/
---------------------
          9

$$\frac{x \sin{\left(\frac{x}{3} \right)} - 6 \cos{\left(\frac{x}{3} \right)}}{9}$$

Simplify

The graph

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Derivative of 3xcos(x/3)-9sin(x/3)

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The solution