Mister Exam

Lang:

The derivative of the function/ x/e^x

Derivative of x/e^x

The solution

You have entered [src]

x 
--
 x
e

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1       -x
-- - x*e  
 x        
e

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

Simplify

The second derivative [src]

          -x
(-2 + x)*e

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

Simplify

The third derivative [src]

         -x
(3 - x)*e

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

Simplify

The graph

Derivative of x/e^x