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Derivative of:
Derivative of x*ctgx
Derivative of x-cos(x)
Derivative of lnx^2
Derivative of ln(x)^2
Identical expressions
x*sin(pix/ two)
x multiply by sinus of ( Pi x divide by 2)
x multiply by sinus of ( Pi x divide by two)
xsin(pix/2)
xsinpix/2
x*sin(pix divide by 2)
Similar expressions
x*sin(pi*x/2^(-2))
x*sin(pi*x/2^(-1))

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

Derivative of x*sin(pix/2)

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \sin{\left(\frac{\pi x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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 is: $\frac{\pi x \cos{\left(\frac{\pi x}{2} \right)}}{2} + \sin{\left(\frac{\pi x}{2} \right)}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   2 /     /pi*x\           /pi*x\\ 
-pi *|6*sin|----| + pi*x*cos|----|| 
     \     \ 2  /           \ 2  // 
------------------------------------
                 8

$$- \frac{\pi^{2} \left(\pi x \cos{\left(\frac{\pi x}{2} \right)} + 6 \sin{\left(\frac{\pi x}{2} \right)}\right)}{8}$$

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of x*sin(pix/2)

The graph:

Piecewise:

The solution