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Derivative of:
Derivative of u
Derivative of (x-1)/2
Derivative of log(e)
Derivative of e^(2*cos(t)^(2)-2*sin(t)^(2))
Identical expressions
sqrt(three)*x*cos(x/ three)
square root of (3) multiply by x multiply by co sinus of e of (x divide by 3)
square root of (three) multiply by x multiply by co sinus of e of (x divide by three)
√(3)*x*cos(x/3)
sqrt(3)xcos(x/3)
sqrt3xcosx/3
sqrt(3)*x*cos(x divide by 3)

Derivative of sqrt(3)xcos(x/3)

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//

$$\frac{d}{d x} \sqrt{3} x \cos{\left(\frac{x}{3} \right)}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \cos{\left(\frac{x}{3} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \frac{x}{3}$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{3}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $\frac{1}{3}$
    The result of the chain rule is:
    
    $- \frac{\sin{\left(\frac{x}{3} \right)}}{3}$
  The result is: $- \frac{x \sin{\left(\frac{x}{3} \right)}}{3} + \cos{\left(\frac{x}{3} \right)}$
So, the result is: $\sqrt{3} \left(- \frac{x \sin{\left(\frac{x}{3} \right)}}{3} + \cos{\left(\frac{x}{3} \right)}\right)$
Now simplify:

$\sqrt{3} \left(- \frac{x \sin{\left(\frac{x}{3} \right)}}{3} + \cos{\left(\frac{x}{3} \right)}\right)$

The answer is:

$\sqrt{3} \left(- \frac{x \sin{\left(\frac{x}{3} \right)}}{3} + \cos{\left(\frac{x}{3} \right)}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   ___    /x\
               x*\/ 3 *sin|-|
  ___    /x\              \3/
\/ 3 *cos|-| - --------------
         \3/         3

$$- \frac{\sqrt{3} x \sin{\left(\frac{x}{3} \right)}}{3} + \sqrt{3} \cos{\left(\frac{x}{3} \right)}$$

Simplify

The second derivative [src]

   ___ /     /x\        /x\\ 
-\/ 3 *|6*sin|-| + x*cos|-|| 
       \     \3/        \3// 
-----------------------------
              9

$$- \frac{\sqrt{3} \left(x \cos{\left(\frac{x}{3} \right)} + 6 \sin{\left(\frac{x}{3} \right)}\right)}{9}$$

Simplify

The third derivative [src]

  ___ /       /x\        /x\\
\/ 3 *|- 9*cos|-| + x*sin|-||
      \       \3/        \3//
-----------------------------
              27

$$\frac{\sqrt{3} \left(x \sin{\left(\frac{x}{3} \right)} - 9 \cos{\left(\frac{x}{3} \right)}\right)}{27}$$

Simplify

The graph

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