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3^(sin2x^3+4sin2x)

Derivative of 3^(sin2x^3+4sin2x)

Function f() - derivative -N order at the point
v

The graph:

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Piecewise:

The solution

You have entered [src]
    3                  
 sin (2*x) + 4*sin(2*x)
3                      
$$3^{\sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}}$$
  /    3                  \
d | sin (2*x) + 4*sin(2*x)|
--\3                      /
dx                         
$$\frac{d}{d x} 3^{\sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}}$$
Detail solution
  1. Let .

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

    1. Differentiate term by term:

      1. Let .

      2. Apply the power rule: goes to

      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:

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

        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:

        So, the result is:

      The result is:

    The result of the chain rule is:

  3. Now simplify:


The answer is:

The graph
The first derivative [src]
    3                                                             
 sin (2*x) + 4*sin(2*x) /                  2              \       
3                      *\8*cos(2*x) + 6*sin (2*x)*cos(2*x)/*log(3)
$$3^{\sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}} \left(6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} + 8 \cos{\left(2 x \right)}\right) \log{\left(3 \right)}$$
The second derivative [src]
   /       2     \          /                                                              2                 \       
   \4 + sin (2*x)/*sin(2*x) |  /         2             2     \            /         2     \     2            |       
4*3                        *\- \4 - 6*cos (2*x) + 3*sin (2*x)/*sin(2*x) + \4 + 3*sin (2*x)/ *cos (2*x)*log(3)/*log(3)
$$4 \cdot 3^{\left(\sin^{2}{\left(2 x \right)} + 4\right) \sin{\left(2 x \right)}} \left(\left(3 \sin^{2}{\left(2 x \right)} + 4\right)^{2} \log{\left(3 \right)} \cos^{2}{\left(2 x \right)} - \left(3 \sin^{2}{\left(2 x \right)} - 6 \cos^{2}{\left(2 x \right)} + 4\right) \sin{\left(2 x \right)}\right) \log{\left(3 \right)}$$
The third derivative [src]
   /       2     \          /                                                   3                                                                                        \                
   \4 + sin (2*x)/*sin(2*x) |           2             2        /         2     \     2         2        /         2     \ /         2             2     \                |                
8*3                        *\-4 - 21*sin (2*x) + 6*cos (2*x) + \4 + 3*sin (2*x)/ *cos (2*x)*log (3) - 3*\4 + 3*sin (2*x)/*\4 - 6*cos (2*x) + 3*sin (2*x)/*log(3)*sin(2*x)/*cos(2*x)*log(3)
$$8 \cdot 3^{\left(\sin^{2}{\left(2 x \right)} + 4\right) \sin{\left(2 x \right)}} \left(\left(3 \sin^{2}{\left(2 x \right)} + 4\right)^{3} \log{\left(3 \right)}^{2} \cos^{2}{\left(2 x \right)} - 3 \cdot \left(3 \sin^{2}{\left(2 x \right)} + 4\right) \left(3 \sin^{2}{\left(2 x \right)} - 6 \cos^{2}{\left(2 x \right)} + 4\right) \log{\left(3 \right)} \sin{\left(2 x \right)} - 21 \sin^{2}{\left(2 x \right)} + 6 \cos^{2}{\left(2 x \right)} - 4\right) \log{\left(3 \right)} \cos{\left(2 x \right)}$$
The graph
Derivative of 3^(sin2x^3+4sin2x)