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

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Derivative of:
Derivative of log(cos(x))
Derivative of pi*sin((pi/3)*t)
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Identical expressions
pi*sin((pi/ three)*t)
Pi multiply by sinus of (( Pi divide by 3) multiply by t)
Pi multiply by sinus of (( Pi divide by three) multiply by t)
pisin((pi/3)t)
pisinpi/3t
pi*sin((pi divide by 3)*t)
Similar expressions
(2pi*sin((pi/3)t))/3

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

Derivative of pisin((pi/3)t)

The solution

You have entered [src]

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   3    /pi*t\ 
-pi *sin|----| 
        \ 3  / 
---------------
       9

$$- \frac{\pi^{3} \sin{\left(\frac{\pi t}{3} \right)}}{9}$$

Simplify

The third derivative [src]

   4    /pi*t\ 
-pi *cos|----| 
        \ 3  / 
---------------
       27

$$- \frac{\pi^{4} \cos{\left(\frac{\pi t}{3} \right)}}{27}$$

Simplify

The graph

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