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Derivative of:
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Identical expressions
log(sin(x/ two))-cot(x/ two)
logarithm of ( sinus of (x divide by 2)) minus cotangent of (x divide by 2)
logarithm of ( sinus of (x divide by two)) minus cotangent of (x divide by two)
logsinx/2-cotx/2
log(sin(x divide by 2))-cot(x divide by 2)
Similar expressions
log(sin(x/2))+cot(x/2)

The derivative of the function/ log(sin(x/2))-cot(x/2)

Derivative of log(sin(x/2))-cot(x/2)

The solution

You have entered [src]

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

\log{\left(\sin{\left(\frac{x}{2} \right)} \right)} - \cot{\left(\frac{x}{2} \right)}

log(sin(x/2)) - cot(x/2)

Detail solution

Differentiate $\log{\left(\sin{\left(\frac{x}{2} \right)} \right)} - \cot{\left(\frac{x}{2} \right)}$ term by term:
1. Let $u = \sin{\left(\frac{x}{2} \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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:
  
  $\frac{\cos{\left(\frac{x}{2} \right)}}{2 \sin{\left(\frac{x}{2} \right)}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  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 $\frac{d}{d x} \tan{\left(\frac{x}{2} \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(\frac{x}{2} \right)} = \frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}$
      
      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 $\frac{d}{d x} f{\left(x \right)}$ :
      
      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:
      
      $- \frac{\sin{\left(\frac{x}{2} \right)}}{2}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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:
      
      $\frac{\cos{\left(\frac{x}{2} \right)}}{2}$
      
      Now plug in to the quotient rule:
      
      $\frac{- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\sin^{2}{\left(\frac{x}{2} \right)}}$
  So, the result is: $\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}$
The result is: $\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{\cos{\left(\frac{x}{2} \right)}}{2 \sin{\left(\frac{x}{2} \right)}}$
Now simplify:

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

The answer is:

\frac{\frac{\sin{\left(x \right)}}{2} + 1}{\left(\cos{\left(x \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of log(sin(x/2))-cot(x/2)

The graph:

Piecewise:

The solution

Method #1

Method #2