Derivative of e^(sqrt(cos2x))

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   __________
 \/ cos(2*x) 
e

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

  /   __________\
d | \/ cos(2*x) |
--\e            /
dx

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

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Detail solution

Let $u = \sqrt{\cos{\left(2 x \right)}}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\cos{\left(2 x \right)}}$ :
1. Let $u = \cos{\left(2 x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x \right)}$ :
  1. Let $u = 2 x$ .
  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 $\frac{d}{d x} 2 x$ :
    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: $2$
    The result of the chain rule is:
    
    $- 2 \sin{\left(2 x \right)}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}$
The result of the chain rule is:

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

The answer is:

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

The graph

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The first derivative [src]

    __________          
  \/ cos(2*x)           
-e            *sin(2*x) 
------------------------
        __________      
      \/ cos(2*x)

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

Simplify

The second derivative [src]

/                      2            2      \    __________
|      __________   sin (2*x)    sin (2*x) |  \/ cos(2*x) 
|- 2*\/ cos(2*x)  + --------- - -----------|*e            
|                    cos(2*x)      3/2     |              
\                               cos   (2*x)/

$$\left(\frac{\sin^{2}{\left(2 x \right)}}{\cos{\left(2 x \right)}} - 2 \sqrt{\cos{\left(2 x \right)}} - \frac{\sin^{2}{\left(2 x \right)}}{\cos^{\frac{3}{2}}{\left(2 x \right)}}\right) e^{\sqrt{\cos{\left(2 x \right)}}}$$

Simplify

The third derivative [src]

/                       2              2             2     \    __________         
|         2          sin (2*x)    3*sin (2*x)   3*sin (2*x)|  \/ cos(2*x)          
|6 - ------------ - ----------- - ----------- + -----------|*e            *sin(2*x)
|      __________      3/2           5/2            2      |                       
\    \/ cos(2*x)    cos   (2*x)   cos   (2*x)    cos (2*x) /

$$\left(\frac{3 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 6 - \frac{\sin^{2}{\left(2 x \right)}}{\cos^{\frac{3}{2}}{\left(2 x \right)}} - \frac{2}{\sqrt{\cos{\left(2 x \right)}}} - \frac{3 \sin^{2}{\left(2 x \right)}}{\cos^{\frac{5}{2}}{\left(2 x \right)}}\right) e^{\sqrt{\cos{\left(2 x \right)}}} \sin{\left(2 x \right)}$$

Simplify

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Derivative of e^(sqrt(cos2x))

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