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Similar expressions
2^(cosx/2)

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

Derivative of 2^(cos(x)/2)

The solution

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 cos(x)
 ------
   2   
2

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

2^(cos(x)/2)

Detail solution

Let $u = \frac{\cos{\left(x \right)}}{2}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\cos{\left(x \right)}}{2}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  So, the result is: $- \frac{\sin{\left(x \right)}}{2}$
The result of the chain rule is:

$- \frac{2^{\frac{\cos{\left(x \right)}}{2}} \log{\left(2 \right)} \sin{\left(x \right)}}{2}$
Now simplify:

$- 2^{\frac{\cos{\left(x \right)}}{2} - 1} \log{\left(2 \right)} \sin{\left(x \right)}$

The answer is:

$- 2^{\frac{\cos{\left(x \right)}}{2} - 1} \log{\left(2 \right)} \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  cos(x)               
  ------               
    2                  
-2      *log(2)*sin(x) 
-----------------------
           2

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

Simplify

The second derivative [src]

 cos(x)                                    
 ------                                    
   2    /               2          \       
2      *\-2*cos(x) + sin (x)*log(2)/*log(2)
-------------------------------------------
                     4

$$\frac{2^{\frac{\cos{\left(x \right)}}{2}} \left(\log{\left(2 \right)} \sin^{2}{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(2 \right)}}{4}$$

Simplify

The third derivative [src]

 cos(x)                                                      
 ------                                                      
   2    /       2       2                     \              
2      *\4 - log (2)*sin (x) + 6*cos(x)*log(2)/*log(2)*sin(x)
-------------------------------------------------------------
                              8

$$\frac{2^{\frac{\cos{\left(x \right)}}{2}} \left(- \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} + 6 \log{\left(2 \right)} \cos{\left(x \right)} + 4\right) \log{\left(2 \right)} \sin{\left(x \right)}}{8}$$

Simplify

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Derivative of 2^(cos(x)/2)

The graph:

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The solution