Mister Exam

Other calculators


5*sin6x-x^2*sin2x

Derivative of 5*sin6x-x^2*sin2x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
              2         
5*sin(6*x) - x *sin(2*x)
$$- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)}$$
d /              2         \
--\5*sin(6*x) - x *sin(2*x)/
dx                          
$$\frac{d}{d x} \left(- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)}\right)$$
Detail solution
  1. Differentiate term by term:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let .

      2. The derivative of sine is cosine:

      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:

      So, the result is:

    2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the product rule:

        ; to find :

        1. Apply the power rule: goes to

        ; to find :

        1. Let .

        2. The derivative of sine is cosine:

        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:

      So, the result is:

    The result is:


The answer is:

The graph
The first derivative [src]
                                2         
30*cos(6*x) - 2*x*sin(2*x) - 2*x *cos(2*x)
$$- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)} + 30 \cos{\left(6 x \right)}$$
The second derivative [src]
  /                                            2         \
2*\-sin(2*x) - 90*sin(6*x) - 4*x*cos(2*x) + 2*x *sin(2*x)/
$$2 \cdot \left(2 x^{2} \sin{\left(2 x \right)} - 4 x \cos{\left(2 x \right)} - \sin{\left(2 x \right)} - 90 \sin{\left(6 x \right)}\right)$$
The third derivative [src]
  /                                2                        \
4*\-270*cos(6*x) - 3*cos(2*x) + 2*x *cos(2*x) + 6*x*sin(2*x)/
$$4 \cdot \left(2 x^{2} \cos{\left(2 x \right)} + 6 x \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)} - 270 \cos{\left(6 x \right)}\right)$$
The graph
Derivative of 5*sin6x-x^2*sin2x