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Derivative of:
Derivative of cos(x/3)
Derivative of √x
Derivative of xcos(x)
Derivative of -x^3+(1/2)*x^2-x+1
Identical expressions
16sin(x/ two)^ two
16 sinus of (x divide by 2) squared
16 sinus of (x divide by two) to the power of two
16sin(x/2)2
16sinx/22
16sin(x/2)²
16sin(x/2) to the power of 2
16sinx/2^2
16sin(x divide by 2)^2

The derivative of the function/ 16sin(x/2)^2

Derivative of 16sin(x/2)^2

The solution

You have entered [src]

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

$$16 \sin^{2}{\left(\frac{x}{2} \right)}$$

16*sin(x/2)^2

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin{\left(\frac{x}{2} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. 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:
  
  $\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$
So, the result is: $16 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$
Now simplify:

$8 \sin{\left(x \right)}$

The answer is:

$8 \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$16 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

       /x\    /x\
-16*cos|-|*sin|-|
       \2/    \2/

$$- 16 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$

Simplify