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3(sin(pix/2)^2)

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Derivative of:
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Derivative of 7x
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Derivative of -(3*pi*cos(pi*t/6))/2
Identical expressions
three (sin(pix/ two)^ two)
3( sinus of ( Pi x divide by 2) squared )
three ( sinus of ( Pi x divide by two) to the power of two)
3(sin(pix/2)2)
3sinpix/22
3(sin(pix/2)²)
3(sin(pix/2) to the power of 2)
3sinpix/2^2
3(sin(pix divide by 2)^2)

The derivative of the function/ 3(sin(pix/2)^2)

Derivative of 3(sin(pix/2)^2)

The solution

You have entered [src]

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

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

d /     2/pi*x\\
--|3*sin |----||
dx\      \ 2  //

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

Transform

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

        /pi*x\    /pi*x\
3*pi*cos|----|*sin|----|
        \ 2  /    \ 2  /

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

Simplify

The second derivative [src]

     2 /   2/pi*x\      2/pi*x\\
-3*pi *|sin |----| - cos |----||
       \    \ 2  /       \ 2  //
--------------------------------
               2

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

Simplify

The third derivative [src]

     3    /pi*x\    /pi*x\
-3*pi *cos|----|*sin|----|
          \ 2  /    \ 2  /

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

Simplify

The graph

Derivative of 3(sin(pix/2)^2)