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Similar expressions
a*sin^3(x/3)

The derivative of the function/ -a*sin^3(x/3)

Derivative of -a*sin^3(x/3)

The solution

You have entered [src]

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

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

(-a)*sin(x/3)^3

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin{\left(\frac{x}{3} \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. 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:
  
  $\sin^{2}{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}$
So, the result is: $- a \sin^{2}{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}$
Now simplify:

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

The answer is:

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

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of -a*sin^3(x/3)

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