Mister Exam

Derivative of 10cos5x*sin5x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
10*cos(5*x)*sin(5*x)
$$\sin{\left(5 x \right)} 10 \cos{\left(5 x \right)}$$
(10*cos(5*x))*sin(5*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. 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:

      So, the result is:

    ; 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:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
        2              2     
- 50*sin (5*x) + 50*cos (5*x)
$$- 50 \sin^{2}{\left(5 x \right)} + 50 \cos^{2}{\left(5 x \right)}$$
The second derivative [src]
-1000*cos(5*x)*sin(5*x)
$$- 1000 \sin{\left(5 x \right)} \cos{\left(5 x \right)}$$
The third derivative [src]
     /   2           2     \
5000*\sin (5*x) - cos (5*x)/
$$5000 \left(\sin^{2}{\left(5 x \right)} - \cos^{2}{\left(5 x \right)}\right)$$