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x^(2/3)*e^(-x)
The derivative of the function/ x^(2/3)*e^(-x)

Derivative of x^(2/3)*e^(-x)

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

You have entered [src] 
 2/3  -x
x   *e
x23e−xx^{\frac{2}{3}} e^{- x} 
d / 2/3  -x\
--\x   *e  /
dx
ddxx23e−x\frac{d}{d x} x^{\frac{2}{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)=x23f{\left(x \right)} = x^{\frac{2}{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: x23x^{\frac{2}{3}} goes to 23x3\frac{2}{3 \sqrt[3]{x}}

    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:

    (−x23ex+2ex3x3)e−2x\left(- x^{\frac{2}{3}} e^{x} + \frac{2 e^{x}}{3 \sqrt[3]{x}}\right) e^{- 2 x}

  2. Now simplify:

    (23−x)e−xx3\frac{\left(\frac{2}{3} - x\right) e^{- x}}{\sqrt[3]{x}}

The answer is:

(23−x)e−xx3\frac{\left(\frac{2}{3} - x\right) e^{- x}}{\sqrt[3]{x}}

The graph
02468-8-6-4-2-10102-1
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                 -x 
   2/3  -x    2*e   
- x   *e   + -------
               3 ___
             3*\/ x
−x23e−x+2e−x3x3- x^{\frac{2}{3}} e^{- x} + \frac{2 e^{- x}}{3 \sqrt[3]{x}}
Simplify
The second derivative [src] 
/ 2/3      4        2   \  -x
|x    - ------- - ------|*e  
|         3 ___      4/3|    
\       3*\/ x    9*x   /
(x23−43x3−29x43)e−x\left(x^{\frac{2}{3}} - \frac{4}{3 \sqrt[3]{x}} - \frac{2}{9 x^{\frac{4}{3}}}\right) e^{- x}
Simplify
The third derivative [src] 
/   2/3     2       2         8   \  -x
|- x    + ----- + ------ + -------|*e  
|         3 ___      4/3       7/3|    
\         \/ x    3*x      27*x   /
(−x23+2x3+23x43+827x73)e−x\left(- x^{\frac{2}{3}} + \frac{2}{\sqrt[3]{x}} + \frac{2}{3 x^{\frac{4}{3}}} + \frac{8}{27 x^{\frac{7}{3}}}\right) e^{- x}
Simplify
The graph
Derivative of x^(2/3)*e^(-x)
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