Mister Exam

Derivative of ctgx/2

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
cot(x)
------
  2   
$$\frac{\cot{\left(x \right)}}{2}$$
d /cot(x)\
--|------|
dx\  2   /
$$\frac{d}{d x} \frac{\cot{\left(x \right)}}{2}$$
Detail solution
  1. 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:

      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. The derivative of sine is cosine:

          To find :

          1. The derivative of cosine is negative sine:

          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. The derivative of cosine is negative sine:

        To find :

        1. The derivative of sine is cosine:

        Now plug in to the quotient rule:

    So, the result is:

  2. Now simplify:


The answer is:

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