Mister Exam

Derivative of ctg4x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
cot(4*x)
$$\cot{\left(4 x \right)}$$
cot(4*x)
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     
-4 - 4*cot (4*x)
$$- 4 \cot^{2}{\left(4 x \right)} - 4$$
The second derivative [src]
   /       2     \         
32*\1 + cot (4*x)/*cot(4*x)
$$32 \left(\cot^{2}{\left(4 x \right)} + 1\right) \cot{\left(4 x \right)}$$
The third derivative [src]
     /       2     \ /         2     \
-128*\1 + cot (4*x)/*\1 + 3*cot (4*x)/
$$- 128 \left(\cot^{2}{\left(4 x \right)} + 1\right) \left(3 \cot^{2}{\left(4 x \right)} + 1\right)$$
The graph
Derivative of ctg4x