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Identical expressions
(exp(x)(sin4x+cos4x))/ five
( exponent of (x)( sinus of 4x plus co sinus of e of 4x)) divide by 5
( exponent of (x)( sinus of 4x plus co sinus of e of 4x)) divide by five
expxsin4x+cos4x/5
(exp(x)(sin4x+cos4x)) divide by 5
Similar expressions
(exp(x)(sin4x-cos4x))/5

The derivative of the function/ (exp(x)(sin4x+cos4x))/5

Derivative of (exp(x)(sin4x+cos4x))/5

The solution

You have entered [src]

 x                      
e *(sin(4*x) + cos(4*x))
------------------------
           5

$$\frac{\left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{x}}{5}$$

  / x                      \
d |e *(sin(4*x) + cos(4*x))|
--|------------------------|
dx\           5            /

$$\frac{d}{d x} \frac{\left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{x}}{5}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin{\left(4 x \right)} + \cos{\left(4 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $\sin{\left(4 x \right)} + \cos{\left(4 x \right)}$ term by term:
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      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: $4$
      The result of the chain rule is:
      
      $4 \cos{\left(4 x \right)}$
    4. Let $u = 4 x$ .
    5. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    6. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
      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: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    The result is: $- 4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}$
  $g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  The result is: $\left(- 4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{x} + \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{x}$
So, the result is: $\frac{\left(- 4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{x}}{5} + \frac{\left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{x}}{5}$
Now simplify:

$\left(- \frac{3 \sin{\left(4 x \right)}}{5} + \cos{\left(4 x \right)}\right) e^{x}$

The answer is:

$\left(- \frac{3 \sin{\left(4 x \right)}}{5} + \cos{\left(4 x \right)}\right) e^{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                            x                          x
(-4*sin(4*x) + 4*cos(4*x))*e    (sin(4*x) + cos(4*x))*e 
----------------------------- + ------------------------
              5                            5

$$\frac{\left(- 4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{x}}{5} + \frac{\left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{x}}{5}$$

Simplify

The second derivative [src]

                             x 
-(7*cos(4*x) + 23*sin(4*x))*e  
-------------------------------
               5

$$- \frac{\left(23 \sin{\left(4 x \right)} + 7 \cos{\left(4 x \right)}\right) e^{x}}{5}$$

Simplify

The third derivative [src]

                             x
(-99*cos(4*x) + 5*sin(4*x))*e 
------------------------------
              5

$$\frac{\left(5 \sin{\left(4 x \right)} - 99 \cos{\left(4 x \right)}\right) e^{x}}{5}$$

Simplify

The graph

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Derivative of (exp(x)(sin4x+cos4x))/5

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The solution