Graph of the Function $y = xe^{-x^2}$

$y = xe^{-x^2}$ is a linear function multiplied by a Gaussian. Since the derivative of the Gaussian $e^{-x^2}$ is $-2xe^{-x^2}$, this function is, up to a constant factor, the slope of the Gaussian. Being odd, it shows one crest and one trough.

Domain and symmetry

The domain is all real numbers. Since $f(-x) = -f(x)$ the function is odd, with point symmetry about the origin. It is positive for $x > 0$ and negative for $x < 0$, and the origin is its only $x$-intercept.

Monotonicity and extrema

The product rule gives the following.

Since $e^{-x^2}$ is positive the sign comes from $1 - 2x^2$, which vanishes at $x = \pm\dfrac{1}{\sqrt{2}}$.

The extreme values are $\pm\dfrac{1}{\sqrt{2e}}$, and the range lies between them. The linear factor stretches with $x$, but the exponential decay overtakes it at once, so the crest never rises very high.

Asymptote

As $x \to \pm\infty$ the decay of $e^{-x^2}$ overwhelms the growth of $x$, so $y \to 0$ and the $x$-axis is a horizontal asymptote. The rational function $\dfrac{x}{1+x^2}$ is likewise odd with a crest and a trough, but it falls off only as $\dfrac{1}{x}$, whereas this one drops to $0$ at a stroke.

Inflection points

The second derivative is $y'' = 2x(2x^2 - 3)e^{-x^2}$. Its sign changes at $x = 0$ and at $x = \pm\sqrt{\dfrac{3}{2}}$, so there are three inflection points.

Relation to the Gaussian

This function is the derivative of the Gaussian multiplied by $-\dfrac{1}{2}$, and something can be read off from that. The inflection points of the Gaussian sit at $x = \pm\dfrac{1}{\sqrt{2}}$, exactly where this function has its extrema. The general rule that the derivative attains an extremum where the original function has an inflection point is made visible here.

Integral

The antiderivative is $-\dfrac{1}{2}e^{-x^2}$.

Being odd, its integral over the whole line is $0$. Restricting to $x \geq 0$ and doubling gives $2xe^{-x^2}$, whose integral is $1$: the density of the Rayleigh distribution¹.

Applications

In image processing, rather than smoothing with a Gaussian and then differentiating, one convolves with this already-differentiated form as a filter. It is the basis of edge detection, from the Canny method onward, and the standard way to obtain a derivative that is robust against noise².

In physics, the distribution of molecular speeds of a two-dimensional gas takes this shape. When each component of the velocity is normally distributed, small speeds have few directions available and large ones are held down by the exponential factor, so a crest forms in between.

1. Rayleigh distribution, Wikipedia
2. Canny edge detector, Wikipedia