Mister Exam

Other calculators

Derivative of sqrt(2)sin(2x)cos(3x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  ___                  
\/ 2 *sin(2*x)*cos(3*x)
$$\sqrt{2} \sin{\left(2 x \right)} \cos{\left(3 x \right)}$$
(sqrt(2)*sin(2*x))*cos(3*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. 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:

    ; 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:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
      ___                         ___                  
- 3*\/ 2 *sin(2*x)*sin(3*x) + 2*\/ 2 *cos(2*x)*cos(3*x)
$$- 3 \sqrt{2} \sin{\left(2 x \right)} \sin{\left(3 x \right)} + 2 \sqrt{2} \cos{\left(2 x \right)} \cos{\left(3 x \right)}$$
The second derivative [src]
   ___                                              
-\/ 2 *(12*cos(2*x)*sin(3*x) + 13*cos(3*x)*sin(2*x))
$$- \sqrt{2} \left(13 \sin{\left(2 x \right)} \cos{\left(3 x \right)} + 12 \sin{\left(3 x \right)} \cos{\left(2 x \right)}\right)$$
The third derivative [src]
  ___                                               
\/ 2 *(-62*cos(2*x)*cos(3*x) + 63*sin(2*x)*sin(3*x))
$$\sqrt{2} \left(63 \sin{\left(2 x \right)} \sin{\left(3 x \right)} - 62 \cos{\left(2 x \right)} \cos{\left(3 x \right)}\right)$$