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x/e^x
The derivative of the function/ x/e^x

Derivative of x/e^x

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
x 
--
 x
e
xex\frac{x}{e^{x}} 
d /x \
--|--|
dx| x|
  \e /
ddxxex\frac{d}{d x} \frac{x}{e^{x}}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=xf{\left(x \right)} = x and g(x)=exg{\left(x \right)} = e^{x}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of exe^{x} is itself.

    Now plug in to the quotient rule:

    (−xex+ex)e−2x\left(- x e^{x} + e^{x}\right) e^{- 2 x}

  2. Now simplify:

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

The answer is:

(1−x)e−x\left(1 - x\right) e^{- x}

The graph
02468-8-6-4-2-1010-500000500000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
1       -x
-- - x*e  
 x        
e
−xe−x+1ex- x e^{- x} + \frac{1}{e^{x}}
Simplify
The second derivative [src] 
          -x
(-2 + x)*e
(x−2)e−x\left(x - 2\right) e^{- x}
Simplify
The third derivative [src] 
         -x
(3 - x)*e
(−x+3)e−x\left(- x + 3\right) e^{- x}
Simplify
The graph
Derivative of x/e^x
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