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Derivative of -sin(5*x)+4*cot(3*x)

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

You have entered [src]
-sin(5*x) + 4*cot(3*x)
$$- \sin{\left(5 x \right)} + 4 \cot{\left(3 x \right)}$$
-sin(5*x) + 4*cot(3*x)
Detail solution
  1. Differentiate term by term:

    1. 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:

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

      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:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
            2                  
-12 - 12*cot (3*x) - 5*cos(5*x)
$$- 5 \cos{\left(5 x \right)} - 12 \cot^{2}{\left(3 x \right)} - 12$$
The second derivative [src]
                 /       2     \         
25*sin(5*x) + 72*\1 + cot (3*x)/*cot(3*x)
$$72 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(3 x \right)} + 25 \sin{\left(5 x \right)}$$
The third derivative [src]
                     2                                               
      /       2     \                          2      /       2     \
- 216*\1 + cot (3*x)/  + 125*cos(5*x) - 432*cot (3*x)*\1 + cot (3*x)/
$$- 216 \left(\cot^{2}{\left(3 x \right)} + 1\right)^{2} - 432 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot^{2}{\left(3 x \right)} + 125 \cos{\left(5 x \right)}$$