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The derivative of the function/ e^(-x)/x

You entered:

e^(-x)/x

What you mean?

1/(e^x*x)

e*(-x)/x

1/(e^x*x)

1/(e^x*x)

Derivative of e^(-x)/x

The solution

You have entered [src]

 -x
e  
---
 x

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

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

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

Transform

Detail solution

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 $g{\left(x \right)} = x e^{x}$ .

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. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  The result is: $x e^{x} + e^{x}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -x    -x
  e     e  
- --- - ---
   x      2
         x

$$- \frac{e^{- x}}{x} - \frac{e^{- x}}{x^{2}}$$

Simplify

The second derivative [src]

/    2   2 \  -x
|1 + - + --|*e  
|    x    2|    
\        x /    
----------------
       x

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

Simplify

The third derivative [src]

 /    3   6    6 \  -x 
-|1 + - + -- + --|*e   
 |    x    3    2|     
 \        x    x /     
-----------------------
           x

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

Simplify

The graph

Derivative of e^(-x)/x