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Similar expressions
1/(3sqrt(2x+sin3x)^2)

The derivative of the function/ 1/(3sqrt(2x-sin3x)^2)

Derivative of 1/(3sqrt(2x-sin3x)^2)

The solution

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          1          
---------------------
                    2
    ________________ 
3*\/ 2*x - sin(3*x)

$$\frac{1}{3 \left(\sqrt{2 x - \sin{\left(3 x \right)}}\right)^{2}}$$

1/(3*(sqrt(2*x - sin(3*x)))^2)

Detail solution

Let $u = 3 \left(\sqrt{2 x - \sin{\left(3 x \right)}}\right)^{2}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 \left(\sqrt{2 x - \sin{\left(3 x \right)}}\right)^{2}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sqrt{2 x - \sin{\left(3 x \right)}}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{2 x - \sin{\left(3 x \right)}}$ :
    1. Let $u = 2 x - \sin{\left(3 x \right)}$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - \sin{\left(3 x \right)}\right)$ :
      1. Differentiate $2 x - \sin{\left(3 x \right)}$ term by term:
        
        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: $2$
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Let $u = 3 x$ .
        
        The derivative of sine is cosine:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
        
        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: $3$
        
        The result of the chain rule is:
        
        $3 \cos{\left(3 x \right)}$
        
        So, the result is: $- 3 \cos{\left(3 x \right)}$
        
        The result is: $2 - 3 \cos{\left(3 x \right)}$
      The result of the chain rule is:
      
      $\frac{2 - 3 \cos{\left(3 x \right)}}{2 \sqrt{2 x - \sin{\left(3 x \right)}}}$
    The result of the chain rule is:
    
    $2 - 3 \cos{\left(3 x \right)}$
  So, the result is: $6 - 9 \cos{\left(3 x \right)}$
The result of the chain rule is:

$- \frac{6 - 9 \cos{\left(3 x \right)}}{9 \left(2 x - \sin{\left(3 x \right)}\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        1                           
------------------*(-6 + 9*cos(3*x))
3*(2*x - sin(3*x))                  
------------------------------------
         3*(2*x - sin(3*x))

$$\frac{\frac{1}{3 \left(2 x - \sin{\left(3 x \right)}\right)} \left(9 \cos{\left(3 x \right)} - 6\right)}{3 \left(2 x - \sin{\left(3 x \right)}\right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                 3                                
              2*(-2 + 3*cos(3*x))    18*(-2 + 3*cos(3*x))*sin(3*x)
-9*cos(3*x) + -------------------- - -----------------------------
                                2           -sin(3*x) + 2*x       
               (-sin(3*x) + 2*x)                                  
------------------------------------------------------------------
                                         2                        
                        (-sin(3*x) + 2*x)

$$\frac{- 9 \cos{\left(3 x \right)} - \frac{18 \left(3 \cos{\left(3 x \right)} - 2\right) \sin{\left(3 x \right)}}{2 x - \sin{\left(3 x \right)}} + \frac{2 \left(3 \cos{\left(3 x \right)} - 2\right)^{3}}{\left(2 x - \sin{\left(3 x \right)}\right)^{2}}}{\left(2 x - \sin{\left(3 x \right)}\right)^{2}}$$

Simplify

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Derivative of 1/(3sqrt(2x-sin3x)^2)

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The solution