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(5x^3+4x+2)*ctan(2x)

Derivative of (5x^3+4x+2)*ctan(2x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
/   3          \         
\5*x  + 4*x + 2/*cot(2*x)
$$\left(5 x^{3} + 4 x + 2\right) \cot{\left(2 x \right)}$$
d //   3          \         \
--\\5*x  + 4*x + 2/*cot(2*x)/
dx                           
$$\frac{d}{d x} \left(5 x^{3} + 4 x + 2\right) \cot{\left(2 x \right)}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Differentiate term by term:

      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:

      2. 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:

      3. The derivative of the constant is zero.

      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]
/          2     \ /   3          \   /        2\         
\-2 - 2*cot (2*x)/*\5*x  + 4*x + 2/ + \4 + 15*x /*cot(2*x)
$$\left(15 x^{2} + 4\right) \cot{\left(2 x \right)} + \left(- 2 \cot^{2}{\left(2 x \right)} - 2\right) \left(5 x^{3} + 4 x + 2\right)$$
The second derivative [src]
  /    /       2     \ /        2\                     /       2     \ /             3\         \
2*\- 2*\1 + cot (2*x)/*\4 + 15*x / + 15*x*cot(2*x) + 4*\1 + cot (2*x)/*\2 + 4*x + 5*x /*cot(2*x)/
$$2 \cdot \left(4 \left(\cot^{2}{\left(2 x \right)} + 1\right) \left(5 x^{3} + 4 x + 2\right) \cot{\left(2 x \right)} + 15 x \cot{\left(2 x \right)} - 2 \cdot \left(15 x^{2} + 4\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)\right)$$
The third derivative [src]
  /                   /       2     \     /       2     \ /         2     \ /             3\      /       2     \ /        2\         \
2*\15*cot(2*x) - 90*x*\1 + cot (2*x)/ - 8*\1 + cot (2*x)/*\1 + 3*cot (2*x)/*\2 + 4*x + 5*x / + 12*\1 + cot (2*x)/*\4 + 15*x /*cot(2*x)/
$$2 \left(12 \cdot \left(15 x^{2} + 4\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} - 8 \left(\cot^{2}{\left(2 x \right)} + 1\right) \left(3 \cot^{2}{\left(2 x \right)} + 1\right) \left(5 x^{3} + 4 x + 2\right) - 90 x \left(\cot^{2}{\left(2 x \right)} + 1\right) + 15 \cot{\left(2 x \right)}\right)$$
The graph
Derivative of (5x^3+4x+2)*ctan(2x)