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Derivative of:
Derivative of x^x^x
Derivative of x^-11
Derivative of (x+5)/(x+1)
Derivative of x^(4/3)
Identical expressions
Аe^(-x)*x^- one
Аe to the power of ( minus x) multiply by x to the power of minus 1
Аe to the power of ( minus x) multiply by x to the power of minus one
Аe(-x)*x-1
Аe-x*x-1
Аe^(-x)x^-1
Аe(-x)x-1
Аe-xx-1
Аe^-xx^-1
Similar expressions
Аe^(x)*x^-1
Аe^(-x)*x^+1

The derivative of the function/ Аe^(-x)*x^-1

Derivative of Аe^(-x)*x^-1

The solution

You have entered [src]

   -x
a*E  
-----
  x

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

(a*E^(-x))/x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $a$ 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:

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

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

The answer is:

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

The first derivative [src]

     -x      -x
  a*e     a*e  
- ----- - -----
    x        2 
            x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify