Mister Exam

Derivative of y=x^3cos2x

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

The graph:

from to

Piecewise:

The solution

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

    ; to find :

    1. Apply the power rule: goes to

    ; 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]
     3               2         
- 2*x *sin(2*x) + 3*x *cos(2*x)
$$- 2 x^{3} \sin{\left(2 x \right)} + 3 x^{2} \cos{\left(2 x \right)}$$
The second derivative [src]
    /                               2         \
2*x*\3*cos(2*x) - 6*x*sin(2*x) - 2*x *cos(2*x)/
$$2 x \left(- 2 x^{2} \cos{\left(2 x \right)} - 6 x \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right)$$
The third derivative [src]
  /                                 2               3         \
2*\3*cos(2*x) - 18*x*sin(2*x) - 18*x *cos(2*x) + 4*x *sin(2*x)/
$$2 \cdot \left(4 x^{3} \sin{\left(2 x \right)} - 18 x^{2} \cos{\left(2 x \right)} - 18 x \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right)$$
The graph
Derivative of y=x^3cos2x