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

Derivative of sin^2(x/2)*ctg(x/2)

The solution

You have entered [src]

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

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

sin(x/2)^2*cot(x/2)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{2}{\left(\frac{x}{2} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$
$g{\left(x \right)} = \cot{\left(\frac{x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\frac{x}{2} \right)} = \frac{1}{\tan{\left(\frac{x}{2} \right)}}$
  2. Let $u = \tan{\left(\frac{x}{2} \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{x}{2} \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(\frac{x}{2} \right)} = \frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      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 $\frac{d}{d x} \frac{x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $\frac{1}{2}$
      
      The result of the chain rule is:
      
      $\frac{\cos{\left(\frac{x}{2} \right)}}{2}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \frac{x}{2}$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $\frac{1}{2}$
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\frac{x}{2} \right)}}{2}$
      
      Now plug in to the quotient rule:
      
      $\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)}}$
    The result of the chain rule is:
    
    $- \frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\frac{x}{2} \right)} = \frac{\cos{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ and $g{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \frac{x}{2}$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $\frac{1}{2}$
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\frac{x}{2} \right)}}{2}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    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 $\frac{d}{d x} \frac{x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      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 plug in to the quotient rule:
    
    $\frac{- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\sin^{2}{\left(\frac{x}{2} \right)}}$
The result is: $- \frac{\left(\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}} + \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}$
Now simplify:

$\frac{\sin{\left(\frac{3 x}{2} \right)} + \sin{\left(\frac{5 x}{2} \right)}}{4 \left(\sin{\left(\frac{x}{2} \right)} + \sin{\left(\frac{3 x}{2} \right)}\right)}$

The answer is:

$\frac{\sin{\left(\frac{3 x}{2} \right)} + \sin{\left(\frac{5 x}{2} \right)}}{4 \left(\sin{\left(\frac{x}{2} \right)} + \sin{\left(\frac{3 x}{2} \right)}\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /         2/x\\                       
        |      cot |-||                       
   2/x\ |  1       \2/|      /x\    /x\    /x\
sin |-|*|- - - -------| + cos|-|*cot|-|*sin|-|
    \2/ \  2      2   /      \2/    \2/    \2/

$$\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) \sin^{2}{\left(\frac{x}{2} \right)} + \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}$$

Simplify

The second derivative [src]

  /   2/x\      2/x\\    /x\      2/x\ /       2/x\\    /x\                              
  |sin |-| - cos |-||*cot|-|   sin |-|*|1 + cot |-||*cot|-|                              
  \    \2/       \2//    \2/       \2/ \        \2//    \2/   /       2/x\\    /x\    /x\
- -------------------------- + ---------------------------- - |1 + cot |-||*cos|-|*sin|-|
              2                             2                 \        \2//    \2/    \2/

$$- \frac{\left(\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}\right) \cot{\left(\frac{x}{2} \right)}}{2} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$

Simplify

The third derivative [src]

  /       2/x\\ /   2/x\      2/x\\                             2/x\ /       2/x\\ /         2/x\\     /       2/x\\    /x\    /x\    /x\
3*|1 + cot |-||*|sin |-| - cos |-||                          sin |-|*|1 + cot |-||*|1 + 3*cot |-||   3*|1 + cot |-||*cos|-|*cot|-|*sin|-|
  \        \2// \    \2/       \2//      /x\    /x\    /x\       \2/ \        \2// \          \2//     \        \2//    \2/    \2/    \2/
----------------------------------- - cos|-|*cot|-|*sin|-| - ------------------------------------- + ------------------------------------
                 4                       \2/    \2/    \2/                     4                                      2

$$\frac{3 \left(\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{4} - \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(3 \cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}{4} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}$$

Simplify

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How to use it?

Identical expressions

Derivative of sin^2(x/2)*ctg(x/2)

The graph:

Piecewise:

The solution

Method #1

Method #2