Mister Exam

Derivative of ctg(x/2)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /x\
cot|-|
   \2/
$$\cot{\left(\frac{x}{2} \right)}$$
cot(x/2)
Detail solution
  1. There are multiple ways to do this derivative.

    Method #1

    1. Rewrite the function to be differentiated:

    2. Let .

    3. Apply the power rule: goes to

    4. Then, apply the chain rule. Multiply by :

      1. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        To find :

        1. Let .

        2. The derivative of sine is cosine:

        3. Then, apply the chain rule. Multiply by :

          1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: goes to

            So, the result is:

          The result of the chain rule is:

        To find :

        1. Let .

        2. The derivative of cosine is negative sine:

        3. Then, apply the chain rule. Multiply by :

          1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: goes to

            So, the result is:

          The result of the chain rule is:

        Now plug in to the quotient rule:

      The result of the chain rule is:

    Method #2

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      To find :

      1. Let .

      2. The derivative of sine is cosine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
         2/x\
      cot |-|
  1       \2/
- - - -------
  2      2   
$$- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}$$
The second derivative [src]
/       2/x\\    /x\
|1 + cot |-||*cot|-|
\        \2//    \2/
--------------------
         2          
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{2}$$
The third derivative [src]
 /       2/x\\ /         2/x\\ 
-|1 + cot |-||*|1 + 3*cot |-|| 
 \        \2// \          \2// 
-------------------------------
               4               
$$- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(3 \cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{4}$$
The graph
Derivative of ctg(x/2)