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sin(t/3)^3

Derivative of sin(t/3)^3

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3/t\
sin |-|
    \3/
$$\sin^{3}{\left(\frac{t}{3} \right)}$$
d /   3/t\\
--|sin |-||
dt\    \3//
$$\frac{d}{d t} \sin^{3}{\left(\frac{t}{3} \right)}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
   2/t\    /t\
sin |-|*cos|-|
    \3/    \3/
$$\sin^{2}{\left(\frac{t}{3} \right)} \cos{\left(\frac{t}{3} \right)}$$
The second derivative [src]
/     2/t\        2/t\\    /t\
|- sin |-| + 2*cos |-||*sin|-|
\      \3/         \3//    \3/
------------------------------
              3               
$$\frac{\left(- \sin^{2}{\left(\frac{t}{3} \right)} + 2 \cos^{2}{\left(\frac{t}{3} \right)}\right) \sin{\left(\frac{t}{3} \right)}}{3}$$
The third derivative [src]
/       2/t\        2/t\\    /t\
|- 7*sin |-| + 2*cos |-||*cos|-|
\        \3/         \3//    \3/
--------------------------------
               9                
$$\frac{\left(- 7 \sin^{2}{\left(\frac{t}{3} \right)} + 2 \cos^{2}{\left(\frac{t}{3} \right)}\right) \cos{\left(\frac{t}{3} \right)}}{9}$$
The graph
Derivative of sin(t/3)^3