Derivative of exp(1/(x-1))

You have entered [src]

     1  
 1*-----
   x - 1
e

$$e^{1 \cdot \frac{1}{x - 1}}$$

  /     1  \
  | 1*-----|
d |   x - 1|
--\e       /
dx

$$\frac{d}{d x} e^{1 \cdot \frac{1}{x - 1}}$$

Transform

Detail solution

Let $u = 1 \cdot \frac{1}{x - 1}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} 1 \cdot \frac{1}{x - 1}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 1$ and $g{\left(x \right)} = x - 1$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of the constant $1$ is zero.
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $- \frac{1}{\left(x - 1\right)^{2}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1   
  ----- 
  x - 1 
-e      
--------
       2
(x - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

3-th derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of exp(1/(x-1))

The graph:

Piecewise:

The solution