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The derivative of the function/ 4sin(pit/2)

Derivative of 4sin(pit/2)

The solution

You have entered [src]

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

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\pi t}{2}$ .
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 $\frac{d}{d t} \frac{\pi t}{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    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: $\pi$
    So, the result is: $\frac{\pi}{2}$
  The result of the chain rule is:
  
  $\frac{\pi \cos{\left(\frac{\pi t}{2} \right)}}{2}$
So, the result is: $2 \pi \cos{\left(\frac{\pi t}{2} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify