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e^(sqrt(cos2x))
The derivative of the function/ e^(sqrt(cos2x))

Derivative of e^(sqrt(cos2x))

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   __________
 \/ cos(2*x) 
e
ecos⁡(2x)e^{\sqrt{\cos{\left(2 x \right)}}} 
  /   __________\
d | \/ cos(2*x) |
--\e            /
dx
ddxecos⁡(2x)\frac{d}{d x} e^{\sqrt{\cos{\left(2 x \right)}}}
Transform
Detail solution

  1. Let u=cos⁡(2x)u = \sqrt{\cos{\left(2 x \right)}}.

  2. The derivative of eue^{u} is itself.

  3. Then, apply the chain rule. Multiply by ddxcos⁡(2x)\frac{d}{d x} \sqrt{\cos{\left(2 x \right)}}:

    1. Let u=cos⁡(2x)u = \cos{\left(2 x \right)}.

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ddxcos⁡(2x)\frac{d}{d x} \cos{\left(2 x \right)}:

      1. Let u=2xu = 2 x.

      2. The derivative of cosine is negative sine:

        dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddx2x\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: xx goes to 11

          So, the result is: 22

        The result of the chain rule is:

        −2sin⁡(2x)- 2 \sin{\left(2 x \right)}

      The result of the chain rule is:

      −sin⁡(2x)cos⁡(2x)- \frac{\sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1010
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Plot the graph f'(x)
The first derivative [src] 
    __________          
  \/ cos(2*x)           
-e            *sin(2*x) 
------------------------
        __________      
      \/ cos(2*x)
−ecos⁡(2x)sin⁡(2x)cos⁡(2x)- \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)/
(sin⁡2(2x)cos⁡(2x)−2cos⁡(2x)−sin⁡2(2x)cos⁡32(2x))ecos⁡(2x)\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) /
(3sin⁡2(2x)cos⁡2(2x)+6−sin⁡2(2x)cos⁡32(2x)−2cos⁡(2x)−3sin⁡2(2x)cos⁡52(2x))ecos⁡(2x)sin⁡(2x)\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
The graph
Derivative of e^(sqrt(cos2x))
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