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Derivative of:
Derivative of x^x^x
Derivative of x^-11
Derivative of (x+5)/(x+1)
Derivative of x^(4/3)
Identical expressions
a*sin^ three (x/ three)
a multiply by sinus of cubed (x divide by 3)
a multiply by sinus of to the power of three (x divide by three)
a*sin3(x/3)
a*sin3x/3
a*sin³(x/3)
a*sin to the power of 3(x/3)
asin^3(x/3)
asin3(x/3)
asin3x/3
asin^3x/3
a*sin^3(x divide by 3)

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

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

The solution

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     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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Identical expressions

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

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The solution