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sin^3x*cos^3x

Derivative of sin^3x*cos^3x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3       3   
sin (x)*cos (x)
$$\sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}$$
d /   3       3   \
--\sin (x)*cos (x)/
dx                 
$$\frac{d}{d x} \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}$$
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. 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       4           4       2   
- 3*cos (x)*sin (x) + 3*cos (x)*sin (x)
$$- 3 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}$$
The second derivative [src]
  /   2    /     2           2   \      2    /   2           2   \        2       2   \              
3*\sin (x)*\- cos (x) + 2*sin (x)/ - cos (x)*\sin (x) - 2*cos (x)/ - 6*cos (x)*sin (x)/*cos(x)*sin(x)
$$3 \left(- \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + \left(2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 6 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$
The third derivative [src]
  /     4    /       2           2   \      4    /       2           2   \        2       2    /   2           2   \        2       2    /     2           2   \\
3*\- cos (x)*\- 2*cos (x) + 7*sin (x)/ - sin (x)*\- 7*cos (x) + 2*sin (x)/ + 9*cos (x)*sin (x)*\sin (x) - 2*cos (x)/ + 9*cos (x)*sin (x)*\- cos (x) + 2*sin (x)//
$$3 \cdot \left(9 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \left(2 \sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin^{4}{\left(x \right)} + 9 \cdot \left(2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{4}{\left(x \right)}\right)$$
The graph
Derivative of sin^3x*cos^3x