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Identical expressions
е^(- one /cosx)
е to the power of ( minus 1 divide by co sinus of e of x)
е to the power of ( minus one divide by co sinus of e of x)
е(-1/cosx)
е-1/cosx
е^-1/cosx
е^(-1 divide by cosx)
Similar expressions
е^(1/cosx)

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

Derivative of е^(-1/cosx)

The solution

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  -1   
 ------
 cos(x)
E

$$e^{- \frac{1}{\cos{\left(x \right)}}}$$

E^(-1/cos(x))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -1           
  ------        
  cos(x)        
-e      *sin(x) 
----------------
       2        
    cos (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)

$$\frac{\left(- \frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - 1\right) e^{- \frac{1}{\cos{\left(x \right)}}}}{\cos{\left(x \right)}}$$

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)

$$\frac{\left(- \frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{6 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - 5 + \frac{3}{\cos{\left(x \right)}}\right) e^{- \frac{1}{\cos{\left(x \right)}}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$

Simplify

The graph

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Derivative of е^(-1/cosx)

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Piecewise:

The solution