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

Derivative of x*e^(-x)

Find the 200th derivative of f(x) = xe−x.
Find the 600th derivative of f(x) = xe−x.

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

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The solution

You have entered [src] 
   -x
x*e
xe−xx e^{- x} 
d /   -x\
--\x*e  /
dx
ddxxe−x\frac{d}{d x} 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
-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.00.51.0-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 -x      -x
e   - x*e
−xe−x+e−x- x e^{- x} + 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
(3−x)e−x\left(3 - x\right) e^{- x}
Simplify
The graph
Derivative of x*e^(-x)
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