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The derivative of the function/ y=1\x-xe^x

Derivative of y=1\x-xe^x

The solution

You have entered [src]

1      x
- - x*E 
x

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

1/x - x*E^x

Detail solution

Differentiate $- e^{x} x + \frac{1}{x}$ term by term:
1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  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: $e^{x} + x e^{x}$
  So, the result is: $- e^{x} - x e^{x}$
The result is: $- x e^{x} - e^{x} - \frac{1}{x^{2}}$

The answer is:

$- x e^{x} - e^{x} - \frac{1}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1     x      x
- -- - e  - x*e 
   2            
  x

$$- x e^{x} - e^{x} - \frac{1}{x^{2}}$$

Simplify

The second derivative [src]

     x   2       x
- 2*e  + -- - x*e 
          3       
         x

$$- x e^{x} - 2 e^{x} + \frac{2}{x^{3}}$$

Simplify

The third derivative [src]

 /   x   6       x\
-|3*e  + -- + x*e |
 |        4       |
 \       x        /

$$- (x e^{x} + 3 e^{x} + \frac{6}{x^{4}})$$

Simplify