Mister Exam

Derivative of ctgx/x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
cot(x)
------
  x   
$$\frac{\cot{\left(x \right)}}{x}$$
cot(x)/x
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    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:

    To find :

    1. Apply the power rule: goes to

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

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