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Derivative of:
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Derivative of x*sin(3*x)
Derivative of cos(3)^(2)*x-sin(3)^(2)*x
Identical expressions
sin(√ three)+ one / three *(sin(three *x))^ two /(cos(six *x))
sinus of (√3) plus 1 divide by 3 multiply by ( sinus of (3 multiply by x)) squared divide by ( co sinus of e of (6 multiply by x))
sinus of (√ three) plus one divide by three multiply by ( sinus of (three multiply by x)) to the power of two divide by ( co sinus of e of (six multiply by x))
sin(√3)+1/3*(sin(3*x))2/(cos(6*x))
sin√3+1/3*sin3*x2/cos6*x
sin(√3)+1/3*(sin(3*x))²/(cos(6*x))
sin(√3)+1/3*(sin(3*x)) to the power of 2/(cos(6*x))
sin(√3)+1/3(sin(3x))^2/(cos(6x))
sin(√3)+1/3(sin(3x))2/(cos(6x))
sin√3+1/3sin3x2/cos6x
sin√3+1/3sin3x^2/cos6x
sin(√3)+1 divide by 3*(sin(3*x))^2 divide by (cos(6*x))
Similar expressions
sin(√3)-1/3*(sin(3*x))^2/(cos(6*x))

The derivative of the function/ sin(√3)+1/3*(sin(3*x))^2/(cos(6*x))

Derivative of sin(√3)+1/3(sin(3x))^2/(cos(6*x))

The solution

You have entered [src]

             /   2     \
             |sin (3*x)|
             |---------|
   /  ___\   \    3    /
sin\\/ 3 / + -----------
               cos(6*x)

$$\frac{\frac{1}{3} \sin^{2}{\left(3 x \right)}}{\cos{\left(6 x \right)}} + \sin{\left(\sqrt{3} \right)}$$

sin(sqrt(3)) + (sin(3*x)^2/3)/cos(6*x)

Detail solution

Differentiate $\frac{\frac{1}{3} \sin^{2}{\left(3 x \right)}}{\cos{\left(6 x \right)}} + \sin{\left(\sqrt{3} \right)}$ term by term:
1. Let $u = \sqrt{3}$ .
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 $\frac{d}{d x} \sqrt{3}$ :
  1. The derivative of the constant $\sqrt{3}$ is zero.
  The result of the chain rule is:
  
  $0$
4. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin^{2}{\left(3 x \right)}$ and $g{\left(x \right)} = 3 \cos{\left(6 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \sin{\left(3 x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
    1. Let $u = 3 x$ .
    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 $\frac{d}{d x} 3 x$ :
      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:
      
      $3 \cos{\left(3 x \right)}$
    The result of the chain rule is:
    
    $6 \sin{\left(3 x \right)} \cos{\left(3 x \right)}$
  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 $u = 6 x$ .
    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 $\frac{d}{d x} 6 x$ :
      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: $6$
      The result of the chain rule is:
      
      $- 6 \sin{\left(6 x \right)}$
    So, the result is: $- 18 \sin{\left(6 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{18 \sin^{2}{\left(3 x \right)} \sin{\left(6 x \right)} + 18 \sin{\left(3 x \right)} \cos{\left(3 x \right)} \cos{\left(6 x \right)}}{9 \cos^{2}{\left(6 x \right)}}$
The result is: $\frac{18 \sin^{2}{\left(3 x \right)} \sin{\left(6 x \right)} + 18 \sin{\left(3 x \right)} \cos{\left(3 x \right)} \cos{\left(6 x \right)}}{9 \cos^{2}{\left(6 x \right)}}$
Now simplify:

$\frac{2 \sin{\left(6 x \right)}}{\cos{\left(12 x \right)} + 1}$

The answer is:

$\frac{2 \sin{\left(6 x \right)}}{\cos{\left(12 x \right)} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                             2         2                                    \
  |   2           2        4*sin (3*x)*sin (6*x)   4*cos(3*x)*sin(3*x)*sin(6*x)|
6*|cos (3*x) + sin (3*x) + --------------------- + ----------------------------|
  |                                 2                        cos(6*x)          |
  \                              cos (6*x)                                     /
--------------------------------------------------------------------------------
                                    cos(6*x)

$$\frac{6 \left(\frac{4 \sin^{2}{\left(3 x \right)} \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + \sin^{2}{\left(3 x \right)} + \frac{4 \sin{\left(3 x \right)} \sin{\left(6 x \right)} \cos{\left(3 x \right)}}{\cos{\left(6 x \right)}} + \cos^{2}{\left(3 x \right)}\right)}{\cos{\left(6 x \right)}}$$

Simplify

The third derivative [src]

   /                           2                      2                       2         3              2                       \
   |                      3*cos (3*x)*sin(6*x)   7*sin (3*x)*sin(6*x)   12*sin (3*x)*sin (6*x)   12*sin (6*x)*cos(3*x)*sin(3*x)|
36*|4*cos(3*x)*sin(3*x) + -------------------- + -------------------- + ---------------------- + ------------------------------|
   |                            cos(6*x)               cos(6*x)                  3                            2                |
   \                                                                          cos (6*x)                    cos (6*x)           /
--------------------------------------------------------------------------------------------------------------------------------
                                                            cos(6*x)

$$\frac{36 \left(\frac{12 \sin^{2}{\left(3 x \right)} \sin^{3}{\left(6 x \right)}}{\cos^{3}{\left(6 x \right)}} + \frac{7 \sin^{2}{\left(3 x \right)} \sin{\left(6 x \right)}}{\cos{\left(6 x \right)}} + \frac{12 \sin{\left(3 x \right)} \sin^{2}{\left(6 x \right)} \cos{\left(3 x \right)}}{\cos^{2}{\left(6 x \right)}} + 4 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{3 \sin{\left(6 x \right)} \cos^{2}{\left(3 x \right)}}{\cos{\left(6 x \right)}}\right)}{\cos{\left(6 x \right)}}$$

Simplify

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Derivative of sin(√3)+1/3(sin(3x))^2/(cos(6*x))

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The solution

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Derivative of sin(√3)+1/3*(sin(3*x))^2/(cos(6*x))

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The solution

Derivative of sin(√3)+1/3(sin(3x))^2/(cos(6*x))