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Derivative of ctg(4x)-8*sin(2x)

Function f() - derivative -N order at the point
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The solution

You have entered [src]
cot(4*x) - 8*sin(2*x)
$$- 8 \sin{\left(2 x \right)} + \cot{\left(4 x \right)}$$
cot(4*x) - 8*sin(2*x)
Detail solution
  1. Differentiate term by term:

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

      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:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                        2     
-4 - 16*cos(2*x) - 4*cot (4*x)
$$- 16 \cos{\left(2 x \right)} - 4 \cot^{2}{\left(4 x \right)} - 4$$
The second derivative [src]
   //       2     \                    \
32*\\1 + cot (4*x)/*cot(4*x) + sin(2*x)/
$$32 \left(\left(\cot^{2}{\left(4 x \right)} + 1\right) \cot{\left(4 x \right)} + \sin{\left(2 x \right)}\right)$$
The third derivative [src]
   /                   2                                         \
   |    /       2     \         2      /       2     \           |
64*\- 2*\1 + cot (4*x)/  - 4*cot (4*x)*\1 + cot (4*x)/ + cos(2*x)/
$$64 \left(- 2 \left(\cot^{2}{\left(4 x \right)} + 1\right)^{2} - 4 \left(\cot^{2}{\left(4 x \right)} + 1\right) \cot^{2}{\left(4 x \right)} + \cos{\left(2 x \right)}\right)$$