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exp(sin(2y))- three *exp(cos(2y))
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exp(sin(2y))-3exp(cos(2y))
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Similar expressions
exp(sin(2y))+3*exp(cos(2y))

The derivative of the function/ exp(sin(2y))-3*exp(cos(2y))

Derivative of exp(sin(2y))-3*exp(cos(2y))

The solution

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 sin(2*y)      cos(2*y)
e         - 3*e

$$e^{\sin{\left(2 y \right)}} - 3 e^{\cos{\left(2 y \right)}}$$

exp(sin(2*y)) - 3*exp(cos(2*y))

Detail solution

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

The answer is:

$2 e^{\sin{\left(2 y \right)}} \cos{\left(2 y \right)} + 6 e^{\cos{\left(2 y \right)}} \sin{\left(2 y \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

            sin(2*y)      cos(2*y)         
2*cos(2*y)*e         + 6*e        *sin(2*y)

$$2 e^{\sin{\left(2 y \right)}} \cos{\left(2 y \right)} + 6 e^{\cos{\left(2 y \right)}} \sin{\left(2 y \right)}$$

Simplify

The second derivative [src]

  /   2       sin(2*y)    sin(2*y)                 2       cos(2*y)               cos(2*y)\
4*\cos (2*y)*e         - e        *sin(2*y) - 3*sin (2*y)*e         + 3*cos(2*y)*e        /

$$4 \left(- e^{\sin{\left(2 y \right)}} \sin{\left(2 y \right)} + e^{\sin{\left(2 y \right)}} \cos^{2}{\left(2 y \right)} - 3 e^{\cos{\left(2 y \right)}} \sin^{2}{\left(2 y \right)} + 3 e^{\cos{\left(2 y \right)}} \cos{\left(2 y \right)}\right)$$

Simplify

The third derivative [src]

  /   3       sin(2*y)             sin(2*y)      cos(2*y)                 3       cos(2*y)               cos(2*y)                        sin(2*y)         \
8*\cos (2*y)*e         - cos(2*y)*e         - 3*e        *sin(2*y) + 3*sin (2*y)*e         - 9*cos(2*y)*e        *sin(2*y) - 3*cos(2*y)*e        *sin(2*y)/

$$8 \left(- 3 e^{\sin{\left(2 y \right)}} \sin{\left(2 y \right)} \cos{\left(2 y \right)} + e^{\sin{\left(2 y \right)}} \cos^{3}{\left(2 y \right)} - e^{\sin{\left(2 y \right)}} \cos{\left(2 y \right)} + 3 e^{\cos{\left(2 y \right)}} \sin^{3}{\left(2 y \right)} - 9 e^{\cos{\left(2 y \right)}} \sin{\left(2 y \right)} \cos{\left(2 y \right)} - 3 e^{\cos{\left(2 y \right)}} \sin{\left(2 y \right)}\right)$$

Simplify

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Derivative of exp(sin(2y))-3*exp(cos(2y))

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The solution