Derivative of sin³(x/3)

You have entered [src]

   3/x\
sin |-|
    \3/

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

sin(x/3)^3

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify