Mister Exam

Derivative of cos^3(2x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3     
cos (2*x)
$$\cos^{3}{\left(2 x \right)}$$
d /   3     \
--\cos (2*x)/
dx           
$$\frac{d}{d x} \cos^{3}{\left(2 x \right)}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

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

    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 of the chain rule is:


The answer is:

The graph
The first derivative [src]
      2              
-6*cos (2*x)*sin(2*x)
$$- 6 \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)}$$
The second derivative [src]
   /     2             2     \         
12*\- cos (2*x) + 2*sin (2*x)/*cos(2*x)
$$12 \cdot \left(2 \sin^{2}{\left(2 x \right)} - \cos^{2}{\left(2 x \right)}\right) \cos{\left(2 x \right)}$$
The third derivative [src]
   /       2             2     \         
24*\- 2*sin (2*x) + 7*cos (2*x)/*sin(2*x)
$$24 \left(- 2 \sin^{2}{\left(2 x \right)} + 7 \cos^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}$$
The graph
Derivative of cos^3(2x)