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Derivative of (sin^2)*(cos3x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   2            
sin (x)*cos(3*x)
$$\sin^{2}{\left(x \right)} \cos{\left(3 x \right)}$$
sin(x)^2*cos(3*x)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

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

      1. The derivative of sine is cosine:

      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                                       
- 3*sin (x)*sin(3*x) + 2*cos(x)*cos(3*x)*sin(x)
$$- 3 \sin^{2}{\left(x \right)} \sin{\left(3 x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(3 x \right)}$$
The second derivative [src]
 /  /   2         2   \                 2                                        \
-\2*\sin (x) - cos (x)/*cos(3*x) + 9*sin (x)*cos(3*x) + 12*cos(x)*sin(x)*sin(3*x)/
$$- (2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(3 x \right)} + 9 \sin^{2}{\left(x \right)} \cos{\left(3 x \right)} + 12 \sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(x \right)})$$
The third derivative [src]
   /   2         2   \                  2                                        
18*\sin (x) - cos (x)/*sin(3*x) + 27*sin (x)*sin(3*x) - 62*cos(x)*cos(3*x)*sin(x)
$$18 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(3 x \right)} + 27 \sin^{2}{\left(x \right)} \sin{\left(3 x \right)} - 62 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(3 x \right)}$$