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

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
     2 
   -x  
   ----
    2  
x*e    
$$x e^{\frac{\left(-1\right) x^{2}}{2}}$$
x*exp((-x^2)/2)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Apply the power rule: goes to

    ; to find :

    1. Let .

    2. The derivative of is itself.

    3. Then, apply the chain rule. Multiply by :

      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: goes to

          So, the result is:

        So, the result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
        2       2 
      -x      -x  
      ----    ----
   2   2       2  
- x *e     + e    
$$- x^{2} e^{\frac{\left(-1\right) x^{2}}{2}} + e^{\frac{\left(-1\right) x^{2}}{2}}$$
The second derivative [src]
               2 
             -x  
             ----
  /      2\   2  
x*\-3 + x /*e    
$$x \left(x^{2} - 3\right) e^{- \frac{x^{2}}{2}}$$
The third derivative [src]
                              2 
                            -x  
                            ----
/        2    2 /      2\\   2  
\-3 + 3*x  - x *\-3 + x //*e    
$$\left(- x^{2} \left(x^{2} - 3\right) + 3 x^{2} - 3\right) e^{- \frac{x^{2}}{2}}$$