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x^(1/3)*e^(-x)
The derivative of the function/ x^(1/3)*e^(-x)

Derivative of x^(1/3)*e^(-x)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
3 ___  -x
\/ x *e
x3e−x\sqrt[3]{x} e^{- x} 
d /3 ___  -x\
--\\/ x *e  /
dx
ddxx3e−x\frac{d}{d x} \sqrt[3]{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)=x3f{\left(x \right)} = \sqrt[3]{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: x3\sqrt[3]{x} goes to 13x23\frac{1}{3 x^{\frac{2}{3}}}

    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+ex3x23)e−2x\left(- \sqrt[3]{x} e^{x} + \frac{e^{x}}{3 x^{\frac{2}{3}}}\right) e^{- 2 x}

  2. Now simplify:

    (13−x)e−xx23\frac{\left(\frac{1}{3} - x\right) e^{- x}}{x^{\frac{2}{3}}}

The answer is:

(13−x)e−xx23\frac{\left(\frac{1}{3} - x\right) e^{- x}}{x^{\frac{2}{3}}}

The graph
02468-8-6-4-2-10101-1
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                -x  
  3 ___  -x    e    
- \/ x *e   + ------
                 2/3
              3*x
−x3e−x+e−x3x23- \sqrt[3]{x} e^{- x} + \frac{e^{- x}}{3 x^{\frac{2}{3}}}
Simplify
The second derivative [src] 
/3 ___     2        2   \  -x
|\/ x  - ------ - ------|*e  
|           2/3      5/3|    
\        3*x      9*x   /
(x3−23x23−29x53)e−x\left(\sqrt[3]{x} - \frac{2}{3 x^{\frac{2}{3}}} - \frac{2}{9 x^{\frac{5}{3}}}\right) e^{- x}
Simplify
The third derivative [src] 
/ 1     3 ___     2         10  \  -x
|---- - \/ x  + ------ + -------|*e  
| 2/3              5/3       8/3|    
\x              3*x      27*x   /
(−x3+1x23+23x53+1027x83)e−x\left(- \sqrt[3]{x} + \frac{1}{x^{\frac{2}{3}}} + \frac{2}{3 x^{\frac{5}{3}}} + \frac{10}{27 x^{\frac{8}{3}}}\right) e^{- x}
Simplify
The graph
Derivative of x^(1/3)*e^(-x)
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