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Identical expressions
exp^(one /(x+ five))
exponent of to the power of (1 divide by (x plus 5))
exponent of to the power of (one divide by (x plus five))
exp(1/(x+5))
exp1/x+5
exp^1/x+5
exp^(1 divide by (x+5))
Similar expressions
exp^(1/(x-5))

The derivative of the function/ exp^(1/(x+5))

Derivative of exp^(1/(x+5))

The solution

You have entered [src]

   1  
 -----
 x + 5
E

$$e^{\frac{1}{x + 5}}$$

E^(1/(x + 5))

Detail solution

Let $u = \frac{1}{x + 5}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{x + 5}$ :
1. Let $u = x + 5$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 5\right)$ :
  1. Differentiate $x + 5$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $5$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $- \frac{1}{\left(x + 5\right)^{2}}$
The result of the chain rule is:

$- \frac{e^{\frac{1}{x + 5}}}{\left(x + 5\right)^{2}}$
Now simplify:

$- \frac{e^{\frac{1}{x + 5}}}{\left(x + 5\right)^{2}}$

The answer is:

$- \frac{e^{\frac{1}{x + 5}}}{\left(x + 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1   
  ----- 
  x + 5 
-e      
--------
       2
(x + 5)

$$- \frac{e^{\frac{1}{x + 5}}}{\left(x + 5\right)^{2}}$$

Simplify

The second derivative [src]

               1  
             -----
/      1  \  5 + x
|2 + -----|*e     
\    5 + x/       
------------------
            3     
     (5 + x)

$$\frac{\left(2 + \frac{1}{x + 5}\right) e^{\frac{1}{x + 5}}}{\left(x + 5\right)^{3}}$$

Simplify

The third derivative [src]

                           1   
                         ----- 
 /       1         6  \  5 + x 
-|6 + -------- + -----|*e      
 |           2   5 + x|        
 \    (5 + x)         /        
-------------------------------
                   4           
            (5 + x)

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

Simplify

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Derivative of exp^(1/(x+5))

The graph:

Piecewise:

The solution