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Derivative of:
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Derivative of 2^x*cos(x)
Identical expressions
two (a*cos2t+sin2t)
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two (a multiply by co sinus of e of 2t plus sinus of 2t)
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Similar expressions
2(a*cos2t-sin2t)

The derivative of the function/ 2(a*cos2t+sin2t)

Derivative of 2(a*cos2t+sin2t)

The solution

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2*(a*cos(2*t) + sin(2*t))

$$2 \left(a \cos{\left(2 t \right)} + \sin{\left(2 t \right)}\right)$$

2*(a*cos(2*t) + sin(2*t))

Detail solution

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

The answer is:

$- 4 a \sin{\left(2 t \right)} + 4 \cos{\left(2 t \right)}$

The first derivative [src]

4*cos(2*t) - 4*a*sin(2*t)

$$- 4 a \sin{\left(2 t \right)} + 4 \cos{\left(2 t \right)}$$

Simplify

The second derivative [src]

-8*(a*cos(2*t) + sin(2*t))

$$- 8 \left(a \cos{\left(2 t \right)} + \sin{\left(2 t \right)}\right)$$

Simplify

The third derivative [src]

16*(-cos(2*t) + a*sin(2*t))

$$16 \left(a \sin{\left(2 t \right)} - \cos{\left(2 t \right)}\right)$$

Simplify