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Derivative of e^2*x
Derivative of cos(x)^2
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Derivative of sin(x^3)
Identical expressions
three ^(sin2x)- one
3 to the power of ( sinus of 2x) minus 1
three to the power of ( sinus of 2x) minus one
3(sin2x)-1
3sin2x-1
3^sin2x-1
Similar expressions
3^(sin2x)+1

The derivative of the function/ 3^(sin2x)-1

Derivative of 3^(sin2x)-1

The solution

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 sin(2*x)    
3         - 1

$$3^{\sin{\left(2 x \right)}} - 1$$

3^sin(2*x) - 1

Detail solution

Differentiate $3^{\sin{\left(2 x \right)}} - 1$ term by term:
1. Let $u = \sin{\left(2 x \right)}$ .
2. $\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$
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.
      1. 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:
  
  $2 \cdot 3^{\sin{\left(2 x \right)}} \log{\left(3 \right)} \cos{\left(2 x \right)}$
4. The derivative of the constant $-1$ is zero.
The result is: $2 \cdot 3^{\sin{\left(2 x \right)}} \log{\left(3 \right)} \cos{\left(2 x \right)}$

The answer is:

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

The graph

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The first derivative [src]

   sin(2*x)                
2*3        *cos(2*x)*log(3)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   sin(2*x) /        2         2                       \                
8*3        *\-1 + cos (2*x)*log (3) - 3*log(3)*sin(2*x)/*cos(2*x)*log(3)

$$8 \cdot 3^{\sin{\left(2 x \right)}} \left(- 3 \log{\left(3 \right)} \sin{\left(2 x \right)} + \log{\left(3 \right)}^{2} \cos^{2}{\left(2 x \right)} - 1\right) \log{\left(3 \right)} \cos{\left(2 x \right)}$$

Simplify

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Derivative of 3^(sin2x)-1

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The solution