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Derivative of:
Derivative of y=4x
Derivative of 3/x^4
Derivative of 2*x+3
Derivative of 2^x+e^x
Identical expressions
4sin((pi/ two)*x)
4 sinus of (( Pi divide by 2) multiply by x)
4 sinus of (( Pi divide by two) multiply by x)
4sin((pi/2)x)
4sinpi/2x
4sin((pi divide by 2)*x)

The derivative of the function/ 4sin((pi/2)*x)

Derivative of 4sin((pi/2)*x)

The solution

You have entered [src]

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

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

d /     /pi*x\\
--|4*sin|----||
dx\     \ 2  //

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

Transform

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of 4sin((pi/2)*x)