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The derivative of the function/ 2*cos(x/2)

Derivative of 2*cos(x/2)

The solution

You have entered [src]

     /x\
2*cos|-|
     \2/

$$2 \cos{\left(\frac{x}{2} \right)}$$

d /     /x\\
--|2*cos|-||
dx\     \2//

$$\frac{d}{d x} 2 \cos{\left(\frac{x}{2} \right)}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{x}{2}$ .
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 $\frac{d}{d x} \frac{x}{2}$ :
  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: $\frac{1}{2}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\frac{x}{2} \right)}}{2}$
So, the result is: $- \sin{\left(\frac{x}{2} \right)}$
Now simplify:

$- \sin{\left(\frac{x}{2} \right)}$

The answer is:

$- \sin{\left(\frac{x}{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /x\
-sin|-|
    \2/

$$- \sin{\left(\frac{x}{2} \right)}$$

Simplify

The second derivative [src]

    /x\ 
-cos|-| 
    \2/ 
--------
   2

$$- \frac{\cos{\left(\frac{x}{2} \right)}}{2}$$

Simplify

The third derivative [src]

   /x\
sin|-|
   \2/
------
  4

$$\frac{\sin{\left(\frac{x}{2} \right)}}{4}$$

Simplify

The graph

Derivative of 2*cos(x/2)