The Gaussian function is defined as follows \begin{equation} \label{eq:univariate.gaussian.func} G(x ; b, I, \mu, \sigma) = b + \frac{I}{\sigma \sqrt{2 \pi}} \exp\left(- \frac{(x - \mu)^2}{2 \sigma^2} \right) \end{equation} where parameter $b \in \mathbb{R}$ denotes the background, that is, the offset from the abscissa. Parameter $I \in \mathbb{R}$ the intensity, the area under the curve to the background $b$, $\mu \in \mathbb{R}$ the mean and $\sigma \in \mathbb{R}$ standard deviation — sometimes also called the width. Note, for $I = 1$ and $b = 0$, term (\ref{eq:univariate.gaussian.func}) is the probability density function of the normal distribution which we denote as $p(x; \mu, \sigma) \widehat{=} G(x; 0, 1, \mu, \sigma)$. For the sake of simplicity we denote $\theta := (b, I, \mu, \sigma)$.

Giving a sample $\left\{x^{(n)}, y^{(n)}\right\}_{n=1}^N$ of size $N$ where $x^{(n)}, y^{(n)}\in \mathbb{R}$ our goal is to minimize the least square error \begin{eqnarray} \label{eq:error.function} E(x) & = & \frac{1}{2 N}\sum_{n=1}^N \left[G(x^{(n)} ; \Theta) - y^{(n)}\right]^2 \end{eqnarray} and thus inferring parameter $\Theta$.

The first order partial derivatives of~(\ref{eq:univariate.gaussian.func}) are \begin{eqnarray} \label{eq:deriv.1d.I} \frac{\partial G(x ; \overline{\theta})}{\partial I} & = & \frac{1}{\sigma \sqrt{2 \pi}} \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right) = p(x; \mu, \sigma)\\ \label{eq:deriv.1d.b} \frac{\partial G(x ; \overline{\theta})}{\partial b} & = & 1 \\ \label{eq:deriv.1d.mu} \frac{\partial G(x ; \overline{\theta})}{\partial \mu} & = & I \, \frac{(x - \mu)}{\sigma^3 \sqrt{2 \pi}} \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2} \right) = I \frac{(x - \mu)}{\sigma^2} \, p(x; \mu, \sigma)\\ \label{eq:deriv.1d.sigma} \frac{\partial G(x ; \overline{\theta})}{\partial \sigma} & = & I \, \frac{((x - \mu)^2 - \sigma^2)}{\sigma^4 \sqrt{2 \pi} } \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2} \right) = I \frac{((x - \mu)^2 - \sigma^2)}{\sigma^3} \, p(x; \mu, \sigma) \end{eqnarray}