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Derivative of:
Derivative of 5e^x
Derivative of 3*x^2*e^x
Derivative of 3*(2-x)^6
Derivative of 1/(2x)
Identical expressions
-pi/ two *sin(pix/ two)
minus Pi divide by 2 multiply by sinus of ( Pi x divide by 2)
minus Pi divide by two multiply by sinus of ( Pi x divide by two)
-pi/2sin(pix/2)
-pi/2sinpix/2
-pi divide by 2*sin(pix divide by 2)
Similar expressions
pi/2*sin(pix/2)

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

Derivative of -pi/2*sin(pix/2)

The solution

You have entered [src]

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

$$\frac{\left(-1\right) \pi}{2} \sin{\left(\frac{\pi x}{2} \right)}$$

((-pi)/2)*sin((pi*x)/2)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\pi x}{2}$ .
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. 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: $\pi$
    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: $- \frac{\pi^{2} \cos{\left(\frac{\pi x}{2} \right)}}{4}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  3    /pi*x\
pi *sin|----|
       \ 2  /
-------------
      8

$$\frac{\pi^{3} \sin{\left(\frac{\pi x}{2} \right)}}{8}$$

Simplify

The third derivative [src]

  4    /pi*x\
pi *cos|----|
       \ 2  /
-------------
      16

$$\frac{\pi^{4} \cos{\left(\frac{\pi x}{2} \right)}}{16}$$

Simplify