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y=e^(-(1/cosx))

Derivative of y=e^(-(1/cosx))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  -1   
 ------
 cos(x)
e      
$$e^{- \frac{1}{\cos{\left(x \right)}}}$$
  /  -1   \
  | ------|
d | cos(x)|
--\e      /
dx         
$$\frac{d}{d x} e^{- \frac{1}{\cos{\left(x \right)}}}$$
Detail solution
  1. Let .

  2. The derivative of is itself.

  3. Then, apply the chain rule. Multiply by :

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

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of cosine is negative sine:

        The result of the chain rule is:

      So, the result is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
   -1           
  ------        
  cos(x)        
-e      *sin(x) 
----------------
       2        
    cos (x)     
$$- \frac{e^{- \frac{1}{\cos{\left(x \right)}}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$
The second derivative [src]
                             -1   
/        2           2   \  ------
|     sin (x)   2*sin (x)|  cos(x)
|-1 + ------- - ---------|*e      
|        3          2    |        
\     cos (x)    cos (x) /        
----------------------------------
              cos(x)              
$$\frac{\left(- \frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - 1\right) e^{- \frac{1}{\cos{\left(x \right)}}}}{\cos{\left(x \right)}}$$
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)                            
$$\frac{\left(- \frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{6 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - 5 + \frac{3}{\cos{\left(x \right)}}\right) e^{- \frac{1}{\cos{\left(x \right)}}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$
The graph
Derivative of y=e^(-(1/cosx))