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Derivative of e^(-kx^2)

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

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The solution

You have entered [src]
     2
 -k*x 
e     
ekx2e^{- k x^{2}}
  /     2\
d | -k*x |
--\e     /
dx        
xekx2\frac{\partial}{\partial x} e^{- k x^{2}}
Detail solution
  1. Let u=kx2u = - k x^{2}.

  2. The derivative of eue^{u} is itself.

  3. Then, apply the chain rule. Multiply by xkx2\frac{\partial}{\partial x} - k x^{2}:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: x2x^{2} goes to 2x2 x

      So, the result is: 2kx- 2 k x

    The result of the chain rule is:

    2kxekx2- 2 k x e^{- k x^{2}}

  4. Now simplify:

    2kxekx2- 2 k x e^{- k x^{2}}


The answer is:

2kxekx2- 2 k x e^{- k x^{2}}

The first derivative [src]
            2
        -k*x 
-2*k*x*e     
2kxekx2- 2 k x e^{- k x^{2}}
The second derivative [src]
                       2
    /          2\  -k*x 
2*k*\-1 + 2*k*x /*e     
2k(2kx21)ekx22 k \left(2 k x^{2} - 1\right) e^{- k x^{2}}
The third derivative [src]
                         2
     2 /         2\  -k*x 
4*x*k *\3 - 2*k*x /*e     
4k2x(2kx2+3)ekx24 k^{2} x \left(- 2 k x^{2} + 3\right) e^{- k x^{2}}