Derivative of 2(cosx-cos2x)

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2*(cos(x) - cos(2*x))

$$2 \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right)$$

2*(cos(x) - cos(2*x))

Detail solution

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

$2 \left(4 \cos{\left(x \right)} - 1\right) \sin{\left(x \right)}$

The answer is:

$2 \left(4 \cos{\left(x \right)} - 1\right) \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

2*(-cos(x) + 4*cos(2*x))

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

Simplify

The third derivative [src]

2*(-8*sin(2*x) + sin(x))

$$2 \left(\sin{\left(x \right)} - 8 \sin{\left(2 x \right)}\right)$$

Simplify