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e^(-(x/ four))sin(2x)
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e^((x/4))sin(2x)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   -x          
            -x     ---         
            ---     4          
             4    e   *sin(2*x)
2*cos(2*x)*e    - -------------
                        4

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                -x 
                                ---
                                 4 
(-488*cos(2*x) + 191*sin(2*x))*e   
-----------------------------------
                 64

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

Simplify

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

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The solution