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Identical expressions
8e^(-x)((-sen(two x)/2)+cos(x))
8e to the power of ( minus x)(( minus sen(2x) divide by 2) plus co sinus of e of (x))
8e to the power of ( minus x)(( minus sen(two x) divide by 2) plus co sinus of e of (x))
8e(-x)((-sen(2x)/2)+cos(x))
8e-x-sen2x/2+cosx
8e^-x-sen2x/2+cosx
8e^(-x)((-sen(2x) divide by 2)+cos(x))
Similar expressions
8e^(-x)((sen(2x)/2)+cos(x))
8e^(x)((-sen(2x)/2)+cos(x))
8e^(-x)((-sen(2x)/2)-cos(x))
8e^(-x)((-sen(2x)/2)+cosx)

The derivative of the function/ 8e^(-x)((-sen(2x)/2)+cos(x))

Derivative of 8e^(-x)((-sen(2x)/2)+cos(x))

The solution

You have entered [src]

   -x /-sin(2*x)          \
8*E  *|---------- + cos(x)|
      \    2              /

$$8 e^{- x} \left(\frac{\left(-1\right) \sin{\left(2 x \right)}}{2} + \cos{\left(x \right)}\right)$$

(8*E^(-x))*((-sin(2*x))/2 + cos(x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - 8 \sin{\left(2 x \right)} + 16 \cos{\left(x \right)}$ and $g{\left(x \right)} = 2 e^{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- 8 \sin{\left(2 x \right)} + 16 \cos{\left(x \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    So, the result is: $- 16 \cos{\left(2 x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $- 16 \sin{\left(x \right)}$
  The result is: $- 16 \sin{\left(x \right)} - 16 \cos{\left(2 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of $e^{x}$ is itself.
  So, the result is: $2 e^{x}$
Now plug in to the quotient rule:

$\frac{\left(2 \left(- 16 \sin{\left(x \right)} - 16 \cos{\left(2 x \right)}\right) e^{x} - 2 \left(- 8 \sin{\left(2 x \right)} + 16 \cos{\left(x \right)}\right) e^{x}\right) e^{- 2 x}}{4}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

    /-sin(2*x)          \  -x                           -x
- 8*|---------- + cos(x)|*e   + 8*(-cos(2*x) - sin(x))*e  
    \    2              /

$$- 8 \left(\frac{\left(-1\right) \sin{\left(2 x \right)}}{2} + \cos{\left(x \right)}\right) e^{- x} + 8 \left(- \sin{\left(x \right)} - \cos{\left(2 x \right)}\right) e^{- x}$$

Simplify

The second derivative [src]

  /                        3*sin(2*x)\  -x
8*|2*cos(2*x) + 2*sin(x) + ----------|*e  
  \                            2     /

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

Simplify

The third derivative [src]

  /                       11*sin(2*x)           \  -x
8*|-2*sin(x) + 2*cos(x) - ----------- + cos(2*x)|*e  
  \                            2                /

$$8 \left(- 2 \sin{\left(x \right)} - \frac{11 \sin{\left(2 x \right)}}{2} + 2 \cos{\left(x \right)} + \cos{\left(2 x \right)}\right) e^{- x}$$

Simplify

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Derivative of 8e^(-x)((-sen(2x)/2)+cos(x))

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The solution