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Derivative of:
Derivative of x×ln(x)
Derivative of x^2*(x-3)
Derivative of (x+2)/(x-2)
Derivative of (x^3-2)*(x^2+1)
Identical expressions
four *sin^ two (pit/ two)
4 multiply by sinus of squared ( Pi t divide by 2)
four multiply by sinus of to the power of two ( Pi t divide by two)
4*sin2(pit/2)
4*sin2pit/2
4*sin²(pit/2)
4*sin to the power of 2(pit/2)
4sin^2(pit/2)
4sin2(pit/2)
4sin2pit/2
4sin^2pit/2
4*sin^2(pit divide by 2)

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

Derivative of 4*sin^2(pit/2)

The solution

You have entered [src]

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

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin{\left(\frac{\pi t}{2} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \sin{\left(\frac{\pi t}{2} \right)}$ :
  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.
        
        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:
    
    $\frac{\pi \cos{\left(\frac{\pi t}{2} \right)}}{2}$
  The result of the chain rule is:
  
  $\pi \sin{\left(\frac{\pi t}{2} \right)} \cos{\left(\frac{\pi t}{2} \right)}$
So, the result is: $4 \pi \sin{\left(\frac{\pi t}{2} \right)} \cos{\left(\frac{\pi t}{2} \right)}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify