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(2pi*sin((pi/3)t))/3

How to use it?
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Identical expressions
(2pi*sin((pi/ three)t))/ three
(2 Pi multiply by sinus of (( Pi divide by 3)t)) divide by 3
(2 Pi multiply by sinus of (( Pi divide by three)t)) divide by three
(2pisin((pi/3)t))/3
2pisinpi/3t/3
(2pi*sin((pi divide by 3)t)) divide by 3

The derivative of the function/ (2pi*sin((pi/3)t))/3

Derivative of (2pi*sin((pi/3)t))/3

The solution

You have entered [src]

        /pi*t\
2*pi*sin|----|
        \ 3  /
--------------
      3

$$\frac{2 \pi \sin{\left(\frac{\pi t}{3} \right)}}{3}$$

  /        /pi*t\\
  |2*pi*sin|----||
d |        \ 3  /|
--|--------------|
dt\      3       /

$$\frac{d}{d t} \frac{2 \pi \sin{\left(\frac{\pi t}{3} \right)}}{3}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\pi t}{3}$ .
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 $\frac{d}{d t} \frac{\pi t}{3}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $t$ goes to $1$
    So, the result is: $\frac{\pi}{3}$
  The result of the chain rule is:
  
  $\frac{\pi \cos{\left(\frac{\pi t}{3} \right)}}{3}$
So, the result is: $\frac{2 \pi^{2} \cos{\left(\frac{\pi t}{3} \right)}}{9}$

The answer is:

$\frac{2 \pi^{2} \cos{\left(\frac{\pi t}{3} \right)}}{9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2    /pi*t\
2*pi *cos|----|
         \ 3  /
---------------
       9

$$\frac{2 \pi^{2} \cos{\left(\frac{\pi t}{3} \right)}}{9}$$

Simplify

The second derivative [src]

     3    /pi*t\
-2*pi *sin|----|
          \ 3  /
----------------
       27

$$- \frac{2 \pi^{3} \sin{\left(\frac{\pi t}{3} \right)}}{27}$$

Simplify

The third derivative [src]

     4    /pi*t\
-2*pi *cos|----|
          \ 3  /
----------------
       81

$$- \frac{2 \pi^{4} \cos{\left(\frac{\pi t}{3} \right)}}{81}$$

Simplify

The graph

Derivative of (2pi*sin((pi/3)t))/3