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three ^(sin2x^ three +4sin2x)
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Similar expressions
3^(sin2x^3-4sin2x)

The derivative of the function/ 3^(sin2x^3+4sin2x)

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

The solution

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    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

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

Transform

Detail solution

Let $u = \sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}$ .
$\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}\right)$ :
1. Differentiate $\sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}$ term by term:
  1. Let $u = \sin{\left(2 x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(2 x \right)}$ :
    1. Let $u = 2 x$ .
    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 $\frac{d}{d x} 2 x$ :
      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: $2$
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    The result of the chain rule is:
    
    $6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 2 x$ .
    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 $\frac{d}{d x} 2 x$ :
      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: $2$
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    So, the result is: $8 \cos{\left(2 x \right)}$
  The result is: $6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} + 8 \cos{\left(2 x \right)}$
The result of the chain rule is:

$3^{\sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}} \left(6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} + 8 \cos{\left(2 x \right)}\right) \log{\left(3 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

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)

$$3^{\sin^{3}{\left(2 x \right)} + 4 \sin{\left(2 x \right)}} \left(6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} + 8 \cos{\left(2 x \right)}\right) \log{\left(3 \right)}$$

Simplify

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)

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

Simplify

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)

$$8 \cdot 3^{\left(\sin^{2}{\left(2 x \right)} + 4\right) \sin{\left(2 x \right)}} \left(\left(3 \sin^{2}{\left(2 x \right)} + 4\right)^{3} \log{\left(3 \right)}^{2} \cos^{2}{\left(2 x \right)} - 3 \cdot \left(3 \sin^{2}{\left(2 x \right)} + 4\right) \left(3 \sin^{2}{\left(2 x \right)} - 6 \cos^{2}{\left(2 x \right)} + 4\right) \log{\left(3 \right)} \sin{\left(2 x \right)} - 21 \sin^{2}{\left(2 x \right)} + 6 \cos^{2}{\left(2 x \right)} - 4\right) \log{\left(3 \right)} \cos{\left(2 x \right)}$$

Simplify

The graph

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

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The solution