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x^3*e^(-x)
The derivative of the function/ x^3*e^(-x)

You entered:

x^3*e^(-x)

What you mean?

x^3/e^x
 
x^((3*e)^(-x))
 
x^((3*e)^(-x))

Derivative of x^3*e^(-x)

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

The graph:

from to

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The solution

You have entered [src] 
 3  -x
x *e
x3e−xx^{3} e^{- x} 
d / 3  -x\
--\x *e  /
dx
ddxx3e−x\frac{d}{d x} x^{3} 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)=x3f{\left(x \right)} = x^{3} 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: x3x^{3} goes to 3x23 x^{2}

    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:

    (−x3ex+3x2ex)e−2x\left(- x^{3} e^{x} + 3 x^{2} e^{x}\right) e^{- 2 x}

  2. Now simplify:

    x2⋅(3−x)e−xx^{2} \cdot \left(3 - x\right) e^{- x}

The answer is:

x2⋅(3−x)e−xx^{2} \cdot \left(3 - x\right) e^{- x}

The graph
02468-8-6-4-2-1010-5000000050000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   3  -x      2  -x
- x *e   + 3*x *e
−x3e−x+3x2e−x- x^{3} e^{- x} + 3 x^{2} e^{- x}
Simplify
The second derivative [src] 
  /     2      \  -x
x*\6 + x  - 6*x/*e
x(x2−6x+6)e−xx \left(x^{2} - 6 x + 6\right) e^{- x}
Simplify
The third derivative [src] 
/     3             2\  -x
\6 - x  - 18*x + 9*x /*e
(−x3+9x2−18x+6)e−x\left(- x^{3} + 9 x^{2} - 18 x + 6\right) e^{- x}
Simplify
The graph
Derivative of x^3*e^(-x)
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