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Limit of the function:
x*exp(-x^2/2)
Identical expressions
x*exp(-x^ two / two)
x multiply by exponent of ( minus x squared divide by 2)
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x*exp(-x2/2)
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xexp(-x^2/2)
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x*exp(-x^2 divide by 2)
Similar expressions
x*exp(x^2/2)

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

Derivative of x*exp(-x^2/2)

The solution

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     2 
   -x  
   ----
    2  
x*e

$$x e^{\frac{\left(-1\right) x^{2}}{2}}$$

x*exp((-x^2)/2)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = e^{\frac{\left(-1\right) x^{2}}{2}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{\left(-1\right) x^{2}}{2}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\left(-1\right) x^{2}}{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $- 2 x$
    So, the result is: $- x$
  The result of the chain rule is:
  
  $- x e^{\frac{\left(-1\right) x^{2}}{2}}$
The result is: $- x^{2} e^{\frac{\left(-1\right) x^{2}}{2}} + e^{\frac{\left(-1\right) x^{2}}{2}}$
Now simplify:

$\left(1 - x^{2}\right) e^{- \frac{x^{2}}{2}}$

The answer is:

$\left(1 - x^{2}\right) e^{- \frac{x^{2}}{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2       2 
      -x      -x  
      ----    ----
   2   2       2  
- x *e     + e

$$- x^{2} e^{\frac{\left(-1\right) x^{2}}{2}} + e^{\frac{\left(-1\right) x^{2}}{2}}$$

Simplify

The second derivative [src]

               2 
             -x  
             ----
  /      2\   2  
x*\-3 + x /*e

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

Simplify

The third derivative [src]

                              2 
                            -x  
                            ----
/        2    2 /      2\\   2  
\-3 + 3*x  - x *\-3 + x //*e

$$\left(- x^{2} \left(x^{2} - 3\right) + 3 x^{2} - 3\right) e^{- \frac{x^{2}}{2}}$$

Simplify

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Derivative of x*exp(-x^2/2)

The graph:

Piecewise:

The solution