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Derivative of:
Derivative of ax*e^(-ax^2)
Derivative of 1/(sinx)
Derivative of sin2xcos²x
Derivative of 4e^(3x)
Identical expressions
ax*e^(-ax^ two)
ax multiply by e to the power of ( minus ax squared )
ax multiply by e to the power of ( minus ax to the power of two)
ax*e(-ax2)
ax*e-ax2
ax*e^(-ax²)
ax*e to the power of (-ax to the power of 2)
axe^(-ax^2)
axe(-ax2)
axe-ax2
axe^-ax^2
Similar expressions
ax*e^(ax^2)

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

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

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

Apply the product rule:

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

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

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

The answer is:

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

The first derivative [src]

       2                2
   -a*x       2  2  -a*x 
a*e      - 2*a *x *e

$$- 2 a^{2} x^{2} e^{- a x^{2}} + a e^{- a x^{2}}$$

Simplify

The second derivative [src]

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

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

Simplify

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

$$2 a^{2} \left(- 2 a x^{2} \left(2 a x^{2} - 3\right) + 6 a x^{2} - 3\right) e^{- a x^{2}}$$

Simplify