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Derivative of cos(x/3)^(3)

You have entered [src]

   3/x\
cos |-|
    \3/

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

cos(x/3)^3

Detail solution

Let $u = \cos{\left(\frac{x}{3} \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(\frac{x}{3} \right)}$ :
1. Let $u = \frac{x}{3}$ .
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 $\frac{d}{d x} \frac{x}{3}$ :
  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: $\frac{1}{3}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\frac{x}{3} \right)}}{3}$
The result of the chain rule is:

$- \sin{\left(\frac{x}{3} \right)} \cos^{2}{\left(\frac{x}{3} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/       2/x\        2/x\\    /x\
|- 2*sin |-| + 7*cos |-||*sin|-|
\        \3/         \3//    \3/
--------------------------------
               9

$$\frac{\left(- 2 \sin^{2}{\left(\frac{x}{3} \right)} + 7 \cos^{2}{\left(\frac{x}{3} \right)}\right) \sin{\left(\frac{x}{3} \right)}}{9}$$

Simplify

The graph