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Derivative of cos(pi*x/3)

You have entered [src]

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

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

cos((pi*x)/3)

Detail solution

Let $u = \frac{\pi x}{3}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
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.
    1. 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:

$- \frac{\pi \sin{\left(\frac{\pi x}{3} \right)}}{3}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify