Mister Exam

Derivative of sin2xcos²x

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 (x)
$$\sin{\left(2 x \right)} \cos^{2}{\left(x \right)}$$
d /            2   \
--\sin(2*x)*cos (x)/
dx                  
$$\frac{d}{d x} \sin{\left(2 x \right)} \cos^{2}{\left(x \right)}$$
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. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of cosine is negative sine:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     2                                       
2*cos (x)*cos(2*x) - 2*cos(x)*sin(x)*sin(2*x)
$$- 2 \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)} + 2 \cos^{2}{\left(x \right)} \cos{\left(2 x \right)}$$
The second derivative [src]
  //   2         2   \                 2                                       \
2*\\sin (x) - cos (x)/*sin(2*x) - 2*cos (x)*sin(2*x) - 4*cos(x)*cos(2*x)*sin(x)/
$$2 \left(- 4 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)} - 2 \sin{\left(2 x \right)} \cos^{2}{\left(x \right)} + \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(2 x \right)}\right)$$
The third derivative [src]
  /       2                 /   2         2   \                                    \
4*\- 2*cos (x)*cos(2*x) + 3*\sin (x) - cos (x)/*cos(2*x) + 8*cos(x)*sin(x)*sin(2*x)/
$$4 \cdot \left(8 \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)} \cos{\left(2 x \right)} + 3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(2 x \right)}\right)$$
The graph
Derivative of sin2xcos²x