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Derivative of:
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Identical expressions
five -2cos(pix/ three)
5 minus 2 co sinus of e of ( Pi x divide by 3)
five minus 2 co sinus of e of ( Pi x divide by three)
5-2cospix/3
5-2cos(pix divide by 3)
Similar expressions
5+2cos(pix/3)

The derivative of the function/ 5-2cos(pix/3)

Derivative of 5-2cos(pix/3)

The solution

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         /pi*x\
5 - 2*cos|----|
         \ 3  /

$$5 - 2 \cos{\left(\frac{\pi x}{3} \right)}$$

5 - 2*cos((pi*x)/3)

Detail solution

Differentiate $5 - 2 \cos{\left(\frac{\pi x}{3} \right)}$ term by term:
1. The derivative of the constant $5$ is zero.
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{\pi x}{3}$ .
  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 $\frac{d}{d x} \frac{\pi x}{3}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      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: $\pi$
      So, the result is: $\frac{\pi}{3}$
    The result of the chain rule is:
    
    $- \frac{\pi \sin{\left(\frac{\pi x}{3} \right)}}{3}$
  So, the result is: $\frac{2 \pi \sin{\left(\frac{\pi x}{3} \right)}}{3}$
The result is: $\frac{2 \pi \sin{\left(\frac{\pi x}{3} \right)}}{3}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

    2    /pi*x\
2*pi *cos|----|
         \ 3  /
---------------
       9

$$\frac{2 \pi^{2} \cos{\left(\frac{\pi x}{3} \right)}}{9}$$

Simplify

The third derivative [src]

     3    /pi*x\
-2*pi *sin|----|
          \ 3  /
----------------
       27

$$- \frac{2 \pi^{3} \sin{\left(\frac{\pi x}{3} \right)}}{27}$$

Simplify