Mister Exam

Derivative of cos(x)*cot(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
cos(x)*cot(x)
$$\cos{\left(x \right)} \cot{\left(x \right)}$$
cos(x)*cot(x)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. The derivative of cosine is negative sine:

    ; 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:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
/        2   \                       
\-1 - cot (x)/*cos(x) - cot(x)*sin(x)
$$\left(- \cot^{2}{\left(x \right)} - 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cot{\left(x \right)}$$
The second derivative [src]
                   /       2   \            /       2   \              
-cos(x)*cot(x) + 2*\1 + cot (x)/*sin(x) + 2*\1 + cot (x)/*cos(x)*cot(x)
$$2 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \cot{\left(x \right)} - \cos{\left(x \right)} \cot{\left(x \right)}$$
The third derivative [src]
                  /       2   \            /       2   \                   /       2   \ /         2   \       
cot(x)*sin(x) + 3*\1 + cot (x)/*cos(x) - 6*\1 + cot (x)/*cot(x)*sin(x) - 2*\1 + cot (x)/*\1 + 3*cot (x)/*cos(x)
$$- 2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - 6 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cot{\left(x \right)} + 3 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} + \sin{\left(x \right)} \cot{\left(x \right)}$$
The graph
Derivative of cos(x)*cot(x)