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Derivative of (cos(x))^sin(x)
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Identical expressions
(cos(sin(3x))/(tg(sqrt(2x+ one))))
( co sinus of e of ( sinus of (3x)) divide by (tg( square root of (2x plus 1))))
( co sinus of e of ( sinus of (3x)) divide by (tg( square root of (2x plus one))))
(cos(sin(3x))/(tg(√(2x+1))))
cossin3x/tgsqrt2x+1
(cos(sin(3x)) divide by (tg(sqrt(2x+1))))
Similar expressions
(cos(sin(3x))/(tg(sqrt(2x-1))))

The derivative of the function/ (cos(sin(3x))/(tg(sqrt(2x+1))))

Derivative of (cos(sin(3x))/(tg(sqrt(2x+1))))

The solution

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 cos(sin(3*x))  
----------------
   /  _________\
tan\\/ 2*x + 1 /

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

cos(sin(3*x))/tan(sqrt(2*x + 1))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \cos{\left(\sin{\left(3 x \right)} \right)}$ and $g{\left(x \right)} = \tan{\left(\sqrt{2 x + 1} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(3 x \right)}$ .
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 $\frac{d}{d x} \sin{\left(3 x \right)}$ :
  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.
      1. 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:
  
  $- 3 \sin{\left(\sin{\left(3 x \right)} \right)} \cos{\left(3 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(\sqrt{2 x + 1} \right)} = \frac{\sin{\left(\sqrt{2 x + 1} \right)}}{\cos{\left(\sqrt{2 x + 1} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(\sqrt{2 x + 1} \right)}$ and $g{\left(x \right)} = \cos{\left(\sqrt{2 x + 1} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \sqrt{2 x + 1}$ .
  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 $\frac{d}{d x} \sqrt{2 x + 1}$ :
    1. Let $u = 2 x + 1$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 1\right)$ :
      1. Differentiate $2 x + 1$ 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 the constant $1$ is zero.
        
        The result is: $2$
      The result of the chain rule is:
      
      $\frac{1}{\sqrt{2 x + 1}}$
    The result of the chain rule is:
    
    $\frac{\cos{\left(\sqrt{2 x + 1} \right)}}{\sqrt{2 x + 1}}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \sqrt{2 x + 1}$ .
  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 $\frac{d}{d x} \sqrt{2 x + 1}$ :
    1. Let $u = 2 x + 1$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 1\right)$ :
      1. Differentiate $2 x + 1$ 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 the constant $1$ is zero.
        
        The result is: $2$
      The result of the chain rule is:
      
      $\frac{1}{\sqrt{2 x + 1}}$
    The result of the chain rule is:
    
    $- \frac{\sin{\left(\sqrt{2 x + 1} \right)}}{\sqrt{2 x + 1}}$
  Now plug in to the quotient rule:
  
  $\frac{\frac{\sin^{2}{\left(\sqrt{2 x + 1} \right)}}{\sqrt{2 x + 1}} + \frac{\cos^{2}{\left(\sqrt{2 x + 1} \right)}}{\sqrt{2 x + 1}}}{\cos^{2}{\left(\sqrt{2 x + 1} \right)}}$
Now plug in to the quotient rule:

$\frac{- \frac{\left(\frac{\sin^{2}{\left(\sqrt{2 x + 1} \right)}}{\sqrt{2 x + 1}} + \frac{\cos^{2}{\left(\sqrt{2 x + 1} \right)}}{\sqrt{2 x + 1}}\right) \cos{\left(\sin{\left(3 x \right)} \right)}}{\cos^{2}{\left(\sqrt{2 x + 1} \right)}} - 3 \sin{\left(\sin{\left(3 x \right)} \right)} \cos{\left(3 x \right)} \tan{\left(\sqrt{2 x + 1} \right)}}{\tan^{2}{\left(\sqrt{2 x + 1} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                             /       2/  _________\\              
  3*cos(3*x)*sin(sin(3*x))   \1 + tan \\/ 2*x + 1 //*cos(sin(3*x))
- ------------------------ - -------------------------------------
         /  _________\             _________    2/  _________\    
      tan\\/ 2*x + 1 /           \/ 2*x + 1 *tan \\/ 2*x + 1 /

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

Simplify

The second derivative [src]

                                                                                 /                                               /       2/  _________\\ \                   /       2/  _________\\                       
       2                                                 /       2/  _________\\ |     2                    1                  2*\1 + tan \\/ 1 + 2*x // |                 6*\1 + tan \\/ 1 + 2*x //*cos(3*x)*sin(sin(3*x))
- 9*cos (3*x)*cos(sin(3*x)) + 9*sin(3*x)*sin(sin(3*x)) + \1 + tan \\/ 1 + 2*x //*|- ------- + ----------------------------- + ---------------------------|*cos(sin(3*x)) + ------------------------------------------------
                                                                                 |  1 + 2*x            3/2    /  _________\                2/  _________\|                             _________    /  _________\          
                                                                                 \            (1 + 2*x)   *tan\\/ 1 + 2*x /   (1 + 2*x)*tan \\/ 1 + 2*x //                           \/ 1 + 2*x *tan\\/ 1 + 2*x /          
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                         /  _________\                                                                                                     
                                                                                                      tan\\/ 1 + 2*x /

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

Simplify

The third derivative [src]

                                                                                                                                                                                                                                                                                                                                                                                                                      /                                               /       2/  _________\\ \                       
                                                                                                                                                                                                                                                                                                                                                                                              /       2/  _________\\ |     2                    1                  2*\1 + tan \\/ 1 + 2*x // |                       
                          /                                                                                                                                          2                                 \                                                                                                                                                                                    9*\1 + tan \\/ 1 + 2*x //*|- ------- + ----------------------------- + ---------------------------|*cos(3*x)*sin(sin(3*x))
                          |                                                                                   /       2/  _________\\         /       2/  _________\\         /       2/  _________\\  |                    /   2                                                              \               /       2/  _________\\ /                            2                   \                             |  1 + 2*x            3/2    /  _________\                2/  _________\|                       
  /       2/  _________\\ |     4                      6                              3                    10*\1 + tan \\/ 1 + 2*x //       6*\1 + tan \\/ 1 + 2*x //       6*\1 + tan \\/ 1 + 2*x //  |                 27*\cos (3*x)*sin(sin(3*x)) + 3*cos(sin(3*x))*sin(3*x) + sin(sin(3*x))/*cos(3*x)   27*\1 + tan \\/ 1 + 2*x //*\sin(3*x)*sin(sin(3*x)) - cos (3*x)*cos(sin(3*x))/                             \            (1 + 2*x)   *tan\\/ 1 + 2*x /   (1 + 2*x)*tan \\/ 1 + 2*x //                       
- \1 + tan \\/ 1 + 2*x //*|------------ - --------------------------- + ------------------------------ - ------------------------------ + ------------------------------ + ----------------------------|*cos(sin(3*x)) + -------------------------------------------------------------------------------- - ----------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------
                          |         3/2            2    /  _________\            5/2    2/  _________\            3/2    2/  _________\            3/2    4/  _________\            2    3/  _________\|                                                    /  _________\                                                             _________    2/  _________\                                                                                   /  _________\                                                     
                          \(1 + 2*x)      (1 + 2*x) *tan\\/ 1 + 2*x /   (1 + 2*x)   *tan \\/ 1 + 2*x /   (1 + 2*x)   *tan \\/ 1 + 2*x /   (1 + 2*x)   *tan \\/ 1 + 2*x /   (1 + 2*x) *tan \\/ 1 + 2*x //                                                 tan\\/ 1 + 2*x /                                                           \/ 1 + 2*x *tan \\/ 1 + 2*x /                                                                                tan\\/ 1 + 2*x /

$$- \frac{9 \left(\tan^{2}{\left(\sqrt{2 x + 1} \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(\sqrt{2 x + 1} \right)} + 1\right)}{\left(2 x + 1\right) \tan^{2}{\left(\sqrt{2 x + 1} \right)}} - \frac{2}{2 x + 1} + \frac{1}{\left(2 x + 1\right)^{\frac{3}{2}} \tan{\left(\sqrt{2 x + 1} \right)}}\right) \sin{\left(\sin{\left(3 x \right)} \right)} \cos{\left(3 x \right)}}{\tan{\left(\sqrt{2 x + 1} \right)}} - \left(\tan^{2}{\left(\sqrt{2 x + 1} \right)} + 1\right) \left(\frac{6 \left(\tan^{2}{\left(\sqrt{2 x + 1} \right)} + 1\right)}{\left(2 x + 1\right)^{2} \tan^{3}{\left(\sqrt{2 x + 1} \right)}} - \frac{6}{\left(2 x + 1\right)^{2} \tan{\left(\sqrt{2 x + 1} \right)}} + \frac{6 \left(\tan^{2}{\left(\sqrt{2 x + 1} \right)} + 1\right)^{2}}{\left(2 x + 1\right)^{\frac{3}{2}} \tan^{4}{\left(\sqrt{2 x + 1} \right)}} - \frac{10 \left(\tan^{2}{\left(\sqrt{2 x + 1} \right)} + 1\right)}{\left(2 x + 1\right)^{\frac{3}{2}} \tan^{2}{\left(\sqrt{2 x + 1} \right)}} + \frac{4}{\left(2 x + 1\right)^{\frac{3}{2}}} + \frac{3}{\left(2 x + 1\right)^{\frac{5}{2}} \tan^{2}{\left(\sqrt{2 x + 1} \right)}}\right) \cos{\left(\sin{\left(3 x \right)} \right)} + \frac{27 \left(3 \sin{\left(3 x \right)} \cos{\left(\sin{\left(3 x \right)} \right)} + \sin{\left(\sin{\left(3 x \right)} \right)} \cos^{2}{\left(3 x \right)} + \sin{\left(\sin{\left(3 x \right)} \right)}\right) \cos{\left(3 x \right)}}{\tan{\left(\sqrt{2 x + 1} \right)}} - \frac{27 \left(\sin{\left(3 x \right)} \sin{\left(\sin{\left(3 x \right)} \right)} - \cos^{2}{\left(3 x \right)} \cos{\left(\sin{\left(3 x \right)} \right)}\right) \left(\tan^{2}{\left(\sqrt{2 x + 1} \right)} + 1\right)}{\sqrt{2 x + 1} \tan^{2}{\left(\sqrt{2 x + 1} \right)}}$$

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Derivative of (cos(sin(3x))/(tg(sqrt(2x+1))))

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