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е^(-1/cosx)
The derivative of the function/ е^(-1/cosx)

Derivative of е^(-1/cosx)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
  -1   
 ------
 cos(x)
E
e−1cos⁡(x)e^{- \frac{1}{\cos{\left(x \right)}}} 
E^(-1/cos(x))
Detail solution

  1. Let u=−1cos⁡(x)u = - \frac{1}{\cos{\left(x \right)}}.

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

  3. Then, apply the chain rule. Multiply by ddx(−1cos⁡(x))\frac{d}{d x} \left(- \frac{1}{\cos{\left(x \right)}}\right):

    1. The derivative of a constant times a function is the constant times the derivative of the function.

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

      2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

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

        1. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        The result of the chain rule is:

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

      So, the result is: −sin⁡(x)cos⁡2(x)- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result of the chain rule is:

    −e−1cos⁡(x)sin⁡(x)cos⁡2(x)- \frac{e^{- \frac{1}{\cos{\left(x \right)}}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The answer is:

−e−1cos⁡(x)sin⁡(x)cos⁡2(x)- \frac{e^{- \frac{1}{\cos{\left(x \right)}}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-500000500000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   -1           
  ------        
  cos(x)        
-e      *sin(x) 
----------------
       2        
    cos (x)
−e−1cos⁡(x)sin⁡(x)cos⁡2(x)- \frac{e^{- \frac{1}{\cos{\left(x \right)}}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}
Simplify
The second derivative [src] 
                             -1   
/        2           2   \  ------
|     sin (x)   2*sin (x)|  cos(x)
|-1 + ------- - ---------|*e      
|        3          2    |        
\     cos (x)    cos (x) /        
----------------------------------
              cos(x)
(−2sin⁡2(x)cos⁡2(x)+sin⁡2(x)cos⁡3(x)−1)e−1cos⁡(x)cos⁡(x)\frac{\left(- \frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - 1\right) e^{- \frac{1}{\cos{\left(x \right)}}}}{\cos{\left(x \right)}}
Simplify
The third derivative [src] 
                                                  -1          
/                 2           2           2   \  ------       
|       3      sin (x)   6*sin (x)   6*sin (x)|  cos(x)       
|-5 + ------ - ------- - --------- + ---------|*e      *sin(x)
|     cos(x)      4          2           3    |               
\              cos (x)    cos (x)     cos (x) /               
--------------------------------------------------------------
                              2                               
                           cos (x)
(−6sin⁡2(x)cos⁡2(x)+6sin⁡2(x)cos⁡3(x)−sin⁡2(x)cos⁡4(x)−5+3cos⁡(x))e−1cos⁡(x)sin⁡(x)cos⁡2(x)\frac{\left(- \frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{6 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - 5 + \frac{3}{\cos{\left(x \right)}}\right) e^{- \frac{1}{\cos{\left(x \right)}}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}
Simplify
The graph
Derivative of е^(-1/cosx)
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