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The derivative of the function/ 2*sin(x-pi/3)*cos(x-pi/3)

Derivative of 2sin(x-pi/3)cos(x-pi/3)

The solution

You have entered [src]

     /    pi\    /    pi\
2*sin|x - --|*cos|x - --|
     \    3 /    \    3 /

$$2 \sin{\left(x - \frac{\pi}{3} \right)} \cos{\left(x - \frac{\pi}{3} \right)}$$

(2*sin(x - pi/3))*cos(x - pi/3)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 2 \sin{\left(x - \frac{\pi}{3} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = x - \frac{\pi}{3}$ .
  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 $\frac{d}{d x} \left(x - \frac{\pi}{3}\right)$ :
    1. Differentiate $x - \frac{\pi}{3}$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $- \frac{\pi}{3}$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $\cos{\left(x - \frac{\pi}{3} \right)}$
  So, the result is: $2 \cos{\left(x - \frac{\pi}{3} \right)}$
$g{\left(x \right)} = \cos{\left(x - \frac{\pi}{3} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - \frac{\pi}{3}$ .
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 $\frac{d}{d x} \left(x - \frac{\pi}{3}\right)$ :
  1. Differentiate $x - \frac{\pi}{3}$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $- \frac{\pi}{3}$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $- \sin{\left(x - \frac{\pi}{3} \right)}$
The result is: $- 2 \sin^{2}{\left(x - \frac{\pi}{3} \right)} + 2 \cos^{2}{\left(x - \frac{\pi}{3} \right)}$
Now simplify:

$- 2 \cos{\left(2 x + \frac{\pi}{3} \right)}$

The answer is:

$- 2 \cos{\left(2 x + \frac{\pi}{3} \right)}$

The graph

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The first derivative [src]

       2/    pi\        2/    pi\
- 2*sin |x - --| + 2*cos |x - --|
        \    3 /         \    3 /

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

Simplify

The second derivative [src]

     /    pi\    /    pi\
8*cos|x + --|*sin|x + --|
     \    6 /    \    6 /

$$8 \sin{\left(x + \frac{\pi}{6} \right)} \cos{\left(x + \frac{\pi}{6} \right)}$$

Simplify

The third derivative [src]

  /   2/    pi\      2/    pi\\
8*|cos |x + --| - sin |x + --||
  \    \    6 /       \    6 //

$$8 \left(- \sin^{2}{\left(x + \frac{\pi}{6} \right)} + \cos^{2}{\left(x + \frac{\pi}{6} \right)}\right)$$

Simplify

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Derivative of 2sin(x-pi/3)cos(x-pi/3)

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The solution

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Derivative of 2*sin(x-pi/3)*cos(x-pi/3)

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Piecewise:

The solution

Derivative of 2sin(x-pi/3)cos(x-pi/3)