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sqrt(3)*x*cos(x/3)

Derivative of sqrt(3)*x*cos(x/3)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  ___      /x\
\/ 3 *x*cos|-|
           \3/
$$\sqrt{3} x \cos{\left(\frac{x}{3} \right)}$$
d /  ___      /x\\
--|\/ 3 *x*cos|-||
dx\           \3//
$$\frac{d}{d x} \sqrt{3} x \cos{\left(\frac{x}{3} \right)}$$
Detail solution
  1. 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 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:

    So, the result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                   ___    /x\
               x*\/ 3 *sin|-|
  ___    /x\              \3/
\/ 3 *cos|-| - --------------
         \3/         3       
$$- \frac{\sqrt{3} x \sin{\left(\frac{x}{3} \right)}}{3} + \sqrt{3} \cos{\left(\frac{x}{3} \right)}$$
The second derivative [src]
   ___ /     /x\        /x\\ 
-\/ 3 *|6*sin|-| + x*cos|-|| 
       \     \3/        \3// 
-----------------------------
              9              
$$- \frac{\sqrt{3} \left(x \cos{\left(\frac{x}{3} \right)} + 6 \sin{\left(\frac{x}{3} \right)}\right)}{9}$$
The third derivative [src]
  ___ /       /x\        /x\\
\/ 3 *|- 9*cos|-| + x*sin|-||
      \       \3/        \3//
-----------------------------
              27             
$$\frac{\sqrt{3} \left(x \sin{\left(\frac{x}{3} \right)} - 9 \cos{\left(\frac{x}{3} \right)}\right)}{27}$$
The graph
Derivative of sqrt(3)*x*cos(x/3)