Derivative of sin(x/2)

You have entered [src]

   /x\
sin|-|
   \2/

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

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

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

Transform

Detail solution

Let $u = \frac{x}{2}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{2}$ :
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: $\frac{1}{2}$
The result of the chain rule is:

$\frac{\cos{\left(\frac{x}{2} \right)}}{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   /x\
cos|-|
   \2/
------
  2

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

    /x\ 
-cos|-| 
    \2/ 
--------
   8

$$- \frac{\cos{\left(\frac{x}{2} \right)}}{8}$$

Simplify

The graph