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Derivative of ax*e^(-ax^2)

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

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Piecewise:

The solution

You have entered [src]
         2
     -a*x 
a*x*E     
$$e^{- a x^{2}} a x$$
(a*x)*E^((-a)*x^2)
Detail solution
  1. Apply the product rule:

    ; to find :

    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:

    ; 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. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The first derivative [src]
       2                2
   -a*x       2  2  -a*x 
a*e      - 2*a *x *e     
$$- 2 a^{2} x^{2} e^{- a x^{2}} + a e^{- a x^{2}}$$
The second derivative [src]
                          2
     2 /          2\  -a*x 
2*x*a *\-3 + 2*a*x /*e     
$$2 a^{2} x \left(2 a x^{2} - 3\right) e^{- a x^{2}}$$
The third derivative [src]
                                               2
   2 /          2        2 /          2\\  -a*x 
2*a *\-3 + 6*a*x  - 2*a*x *\-3 + 2*a*x //*e     
$$2 a^{2} \left(- 2 a x^{2} \left(2 a x^{2} - 3\right) + 6 a x^{2} - 3\right) e^{- a x^{2}}$$