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Derivative of (6x^8)*cot(5x)

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The solution

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   8         
6*x *cot(5*x)
$$6 x^{8} \cot{\left(5 x \right)}$$
(6*x^8)*cot(5*x)
Detail solution
  1. Apply the product rule:

    ; to find :

    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:

    ; 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. 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:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
   8 /          2     \       7         
6*x *\-5 - 5*cot (5*x)/ + 48*x *cot(5*x)
$$6 x^{8} \left(- 5 \cot^{2}{\left(5 x \right)} - 5\right) + 48 x^{7} \cot{\left(5 x \right)}$$
The second derivative [src]
    6 /                   /       2     \       2 /       2     \         \
12*x *\28*cot(5*x) - 40*x*\1 + cot (5*x)/ + 25*x *\1 + cot (5*x)/*cot(5*x)/
$$12 x^{6} \left(25 x^{2} \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} - 40 x \left(\cot^{2}{\left(5 x \right)} + 1\right) + 28 \cot{\left(5 x \right)}\right)$$
The third derivative [src]
    5 /                     /       2     \        3 /       2     \ /         2     \        2 /       2     \         \
12*x *\168*cot(5*x) - 420*x*\1 + cot (5*x)/ - 125*x *\1 + cot (5*x)/*\1 + 3*cot (5*x)/ + 600*x *\1 + cot (5*x)/*cot(5*x)/
$$12 x^{5} \left(- 125 x^{3} \left(\cot^{2}{\left(5 x \right)} + 1\right) \left(3 \cot^{2}{\left(5 x \right)} + 1\right) + 600 x^{2} \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} - 420 x \left(\cot^{2}{\left(5 x \right)} + 1\right) + 168 \cot{\left(5 x \right)}\right)$$