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

Function f() - derivative -N order at the point
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Piecewise:

The solution

You have entered [src]
 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
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Let .

    2. The derivative of cosine is negative sine:

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. The derivative of sine is cosine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      The result of the chain rule is:

    To find :

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. Let .

      2. The derivative of sine is cosine:

      3. Then, apply the chain rule. Multiply by :

        1. Let .

        2. Apply the power rule: goes to

        3. Then, apply the chain rule. Multiply by :

          1. Differentiate term by term:

            1. The derivative of a constant times a function is the constant times the derivative of the function.

              1. Apply the power rule: goes to

              So, the result is:

            2. The derivative of the constant is zero.

            The result is:

          The result of the chain rule is:

        The result of the chain rule is:

      To find :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. Then, apply the chain rule. Multiply by :

        1. Let .

        2. Apply the power rule: goes to

        3. Then, apply the chain rule. Multiply by :

          1. Differentiate term by term:

            1. The derivative of a constant times a function is the constant times the derivative of the function.

              1. Apply the power rule: goes to

              So, the result is:

            2. The derivative of the constant is zero.

            The result is:

          The result of the chain rule is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

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