Mister Exam

Derivative of sin^4xcos^4x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   4       4   
sin (x)*cos (x)
$$\sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}$$
sin(x)^4*cos(x)^4
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]
       3       5           5       3   
- 4*cos (x)*sin (x) + 4*cos (x)*sin (x)
$$- 4 \sin^{5}{\left(x \right)} \cos^{3}{\left(x \right)} + 4 \sin^{3}{\left(x \right)} \cos^{5}{\left(x \right)}$$
The second derivative [src]
     2       2    /   2    /     2           2   \      2    /   2           2   \        2       2   \
4*cos (x)*sin (x)*\sin (x)*\- cos (x) + 3*sin (x)/ - cos (x)*\sin (x) - 3*cos (x)/ - 8*cos (x)*sin (x)/
$$4 \left(- \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + \left(3 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 8 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}$$
The third derivative [src]
  /     4    /       2           2   \      4    /       2           2   \        2       2    /   2           2   \        2       2    /     2           2   \\              
8*\- cos (x)*\- 3*cos (x) + 5*sin (x)/ - sin (x)*\- 5*cos (x) + 3*sin (x)/ + 6*cos (x)*sin (x)*\sin (x) - 3*cos (x)/ + 6*cos (x)*sin (x)*\- cos (x) + 3*sin (x)//*cos(x)*sin(x)
$$8 \left(6 \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \left(3 \sin^{2}{\left(x \right)} - 5 \cos^{2}{\left(x \right)}\right) \sin^{4}{\left(x \right)} + 6 \left(3 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \left(5 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos^{4}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$
The graph
Derivative of sin^4xcos^4x