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Derivative of:
Derivative of x/8
Derivative of 2*x-3
Derivative of x+log(x)
Derivative of x^sqrt(x)
Identical expressions
sin^ two (x/ two)*ctg(x/ two)
sinus of squared (x divide by 2) multiply by ctg(x divide by 2)
sinus of to the power of two (x divide by two) multiply by ctg(x divide by two)
sin2(x/2)*ctg(x/2)
sin2x/2*ctgx/2
sin²(x/2)*ctg(x/2)
sin to the power of 2(x/2)*ctg(x/2)
sin^2(x/2)ctg(x/2)
sin2(x/2)ctg(x/2)
sin2x/2ctgx/2
sin^2x/2ctgx/2
sin^2(x divide by 2)*ctg(x divide by 2)

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

Other calculators

How to use it?

Identical expressions

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

The graph:

Piecewise:

The solution

Method #1

Method #2