Derivative of sin(pi*t/2)

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

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

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

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

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Detail solution

Let $u = \frac{\pi t}{2}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d t} \frac{\pi t}{2}$ :
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: $\frac{\pi}{2}$
The result of the chain rule is:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

   2    /pi*t\ 
-pi *sin|----| 
        \ 2  / 
---------------
       4

$$- \frac{\pi^{2} \sin{\left(\frac{\pi t}{2} \right)}}{4}$$

Simplify

The third derivative [src]

   3    /pi*t\ 
-pi *cos|----| 
        \ 2  / 
---------------
       8

$$- \frac{\pi^{3} \cos{\left(\frac{\pi t}{2} \right)}}{8}$$

Simplify

The graph