Mister Exam

Derivative of 4sin2x+3ctg5x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
4*sin(2*x) + 3*cot(5*x)
$$4 \sin{\left(2 x \right)} + 3 \cot{\left(5 x \right)}$$
4*sin(2*x) + 3*cot(5*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                  
-15 - 15*cot (5*x) + 8*cos(2*x)
$$8 \cos{\left(2 x \right)} - 15 \cot^{2}{\left(5 x \right)} - 15$$
The second derivative [src]
  /                 /       2     \         \
2*\-8*sin(2*x) + 75*\1 + cot (5*x)/*cot(5*x)/
$$2 \left(75 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} - 8 \sin{\left(2 x \right)}\right)$$
The third derivative [src]
   /                                 2                                \
   |                  /       2     \           2      /       2     \|
-2*\16*cos(2*x) + 375*\1 + cot (5*x)/  + 750*cot (5*x)*\1 + cot (5*x)//
$$- 2 \left(375 \left(\cot^{2}{\left(5 x \right)} + 1\right)^{2} + 750 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot^{2}{\left(5 x \right)} + 16 \cos{\left(2 x \right)}\right)$$