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Identical expressions
x^(two / three)*e^(-x)
x to the power of (2 divide by 3) multiply by e to the power of ( minus x)
x to the power of (two divide by three) multiply by e to the power of ( minus x)
x(2/3)*e(-x)
x2/3*e-x
x^(2/3)e^(-x)
x(2/3)e(-x)
x2/3e-x
x^2/3e^-x
x^(2 divide by 3)*e^(-x)
Similar expressions
x^(2/3)*e^(x)

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

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

The solution

You have entered [src]

 2/3  -x
x   *e

$$x^{\frac{2}{3}} e^{- x}$$

d / 2/3  -x\
--\x   *e  /
dx

$$\frac{d}{d x} x^{\frac{2}{3}} e^{- x}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{\frac{2}{3}}$ and $g{\left(x \right)} = e^{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{\frac{2}{3}}$ goes to $\frac{2}{3 \sqrt[3]{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
Now plug in to the quotient rule:

$\left(- x^{\frac{2}{3}} e^{x} + \frac{2 e^{x}}{3 \sqrt[3]{x}}\right) e^{- 2 x}$
Now simplify:

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

The answer is:

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

The graph

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

$$- x^{\frac{2}{3}} e^{- x} + \frac{2 e^{- x}}{3 \sqrt[3]{x}}$$

Simplify

The second derivative [src]

/ 2/3      4        2   \  -x
|x    - ------- - ------|*e  
|         3 ___      4/3|    
\       3*\/ x    9*x   /

$$\left(x^{\frac{2}{3}} - \frac{4}{3 \sqrt[3]{x}} - \frac{2}{9 x^{\frac{4}{3}}}\right) e^{- x}$$

Simplify

The third derivative [src]

/   2/3     2       2         8   \  -x
|- x    + ----- + ------ + -------|*e  
|         3 ___      4/3       7/3|    
\         \/ x    3*x      27*x   /

$$\left(- x^{\frac{2}{3}} + \frac{2}{\sqrt[3]{x}} + \frac{2}{3 x^{\frac{4}{3}}} + \frac{8}{27 x^{\frac{7}{3}}}\right) e^{- x}$$

Simplify

The graph

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

The graph:

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The solution