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Similar expressions
y=e^(2x)sin((x)/(4))

The derivative of the function/ y=e^(-2x)sin((x)/(4))

Derivative of y=e^(-2x)sin((x)/(4))

The solution

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 -2*x    /x\
E    *sin|-|
         \4/

$$e^{- 2 x} \sin{\left(\frac{x}{4} \right)}$$

E^(-2*x)*sin(x/4)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(\frac{x}{4} \right)}$ and $g{\left(x \right)} = e^{2 x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{x}{4}$ .
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 $\frac{d}{d x} \frac{x}{4}$ :
  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: $\frac{1}{4}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(\frac{x}{4} \right)}}{4}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of $e^{u}$ is itself.
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:
  
  $2 e^{2 x}$
Now plug in to the quotient rule:

$\left(- 2 e^{2 x} \sin{\left(\frac{x}{4} \right)} + \frac{e^{2 x} \cos{\left(\frac{x}{4} \right)}}{4}\right) e^{- 4 x}$
Now simplify:

$\frac{\left(- 8 \sin{\left(\frac{x}{4} \right)} + \cos{\left(\frac{x}{4} \right)}\right) e^{- 2 x}}{4}$

The answer is:

$\frac{\left(- 8 \sin{\left(\frac{x}{4} \right)} + \cos{\left(\frac{x}{4} \right)}\right) e^{- 2 x}}{4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      /x\  -2*x
                   cos|-|*e    
     -2*x    /x\      \4/      
- 2*e    *sin|-| + ------------
             \4/        4

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

Simplify

The second derivative [src]

/                 /x\\      
|           63*sin|-||      
|     /x\         \4/|  -2*x
|- cos|-| + ---------|*e    
\     \4/       16   /

$$\left(\frac{63 \sin{\left(\frac{x}{4} \right)}}{16} - \cos{\left(\frac{x}{4} \right)}\right) e^{- 2 x}$$

Simplify

The third derivative [src]

/         /x\          /x\\  -2*x
|- 488*sin|-| + 191*cos|-||*e    
\         \4/          \4//      
---------------------------------
                64

$$\frac{\left(- 488 \sin{\left(\frac{x}{4} \right)} + 191 \cos{\left(\frac{x}{4} \right)}\right) e^{- 2 x}}{64}$$

Simplify

The graph

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Derivative of y=e^(-2x)sin((x)/(4))

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The solution