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Derivative of:
Derivative of sqrt(16-x^2)
Derivative of e^(-kx^2)
Derivative of 3cos3x
Derivative of sqrt(5x-1)
Identical expressions
e^(-kx^ two)
e to the power of ( minus kx squared )
e to the power of ( minus kx to the power of two)
e(-kx2)
e-kx2
e^(-kx²)
e to the power of (-kx to the power of 2)
e^-kx^2
Similar expressions
e^(kx^2)

The derivative of the function/ e^(-kx^2)

Derivative of e^(-kx^2)

The solution

You have entered [src]

     2
 -k*x 
e

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

  /     2\
d | -k*x |
--\e     /
dx

$$\frac{\partial}{\partial x} e^{- k x^{2}}$$

Transform

Detail solution

Let $u = - k x^{2}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} - k x^{2}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  So, the result is: $- 2 k x$
The result of the chain rule is:

$- 2 k x e^{- k x^{2}}$
Now simplify:

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

The answer is:

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

The first derivative [src]

            2
        -k*x 
-2*k*x*e

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

Simplify

The second derivative [src]

                       2
    /          2\  -k*x 
2*k*\-1 + 2*k*x /*e

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

Simplify

The third derivative [src]

                         2
     2 /         2\  -k*x 
4*x*k *\3 - 2*k*x /*e

$$4 k^{2} x \left(- 2 k x^{2} + 3\right) e^{- k x^{2}}$$

Simplify