Mister Exam

Derivative of y=-x^2cos9x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  2         
-x *cos(9*x)
$$- x^{2} \cos{\left(9 x \right)}$$
(-x^2)*cos(9*x)
Detail solution
  1. Apply the product rule:

    ; to find :

    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:

    ; 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         
-2*x*cos(9*x) + 9*x *sin(9*x)
$$9 x^{2} \sin{\left(9 x \right)} - 2 x \cos{\left(9 x \right)}$$
The second derivative [src]
                                  2         
-2*cos(9*x) + 36*x*sin(9*x) + 81*x *cos(9*x)
$$81 x^{2} \cos{\left(9 x \right)} + 36 x \sin{\left(9 x \right)} - 2 \cos{\left(9 x \right)}$$
The third derivative [src]
   /                 2                         \
27*\2*sin(9*x) - 27*x *sin(9*x) + 18*x*cos(9*x)/
$$27 \left(- 27 x^{2} \sin{\left(9 x \right)} + 18 x \cos{\left(9 x \right)} + 2 \sin{\left(9 x \right)}\right)$$
The graph
Derivative of y=-x^2cos9x