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Derivative of:
Derivative of f(x)=x
Derivative of sin^2(x/2)
Derivative of 2x+1
Derivative of 2*cos(3*x)
Integral of d{x}:
sin^2(x/2)
Identical expressions
sin^ two (x/ two)
sinus of squared (x divide by 2)
sinus of to the power of two (x divide by two)
sin2(x/2)
sin2x/2
sin²(x/2)
sin to the power of 2(x/2)
sin^2x/2
sin^2(x divide by 2)

The derivative of the function/ sin^2(x/2)

Derivative of sin^2(x/2)

The solution

You have entered [src]

   2/x\
sin |-|
    \2/

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

sin(x/2)^2

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of sin^2(x/2)