Mister Exam

Derivative of y=sin(4x)cos(2x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
sin(4*x)*cos(2*x)
$$\sin{\left(4 x \right)} \cos{\left(2 x \right)}$$
sin(4*x)*cos(2*x)
Detail solution
  1. Apply the product rule:

    ; 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:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
-2*sin(2*x)*sin(4*x) + 4*cos(2*x)*cos(4*x)
$$- 2 \sin{\left(2 x \right)} \sin{\left(4 x \right)} + 4 \cos{\left(2 x \right)} \cos{\left(4 x \right)}$$
The second derivative [src]
-4*(4*cos(4*x)*sin(2*x) + 5*cos(2*x)*sin(4*x))
$$- 4 \left(4 \sin{\left(2 x \right)} \cos{\left(4 x \right)} + 5 \sin{\left(4 x \right)} \cos{\left(2 x \right)}\right)$$
The third derivative [src]
8*(-14*cos(2*x)*cos(4*x) + 13*sin(2*x)*sin(4*x))
$$8 \left(13 \sin{\left(2 x \right)} \sin{\left(4 x \right)} - 14 \cos{\left(2 x \right)} \cos{\left(4 x \right)}\right)$$
The graph
Derivative of y=sin(4x)cos(2x)