Perturbation Methods

If we have a function, \(f(\mathbf{x})\) of some vector variable, \(\mathbf{x}\), then perturbing \(\mathbf{x}\) slightly from its nominal value, \(\bar{\mathbf{x}}\), by an amount \(\delta \mathbf{x}\) will result a change in the function. Consider a standard Taylor series expansion of \(f\) around its nominal value:

\[ f(\mathbf{x}) \approx \left. f(\mathbf{x}) \right|_{\mathbf{x} = \bar{\mathbf{x}}} + \left. \nabla^T f(\mathbf{x}) \right|_{\mathbf{x} = \bar{\mathbf{x}}} \left( \mathbf{x} - \bar{\mathbf{x}} \right) + \frac{1}{2} \left( \mathbf{x} - \bar{\mathbf{x}} \right)^T \nabla^2 \left. f(\mathbf{x}) \right|_{\mathbf{x} = \bar{\mathbf{x}}} \left(\mathbf{x} - \bar{\mathbf{x}} \right) + \ldots. \]

For convergence, we need \(\mathbf{x} - \bar{\mathbf{x}}\) to be small in magnitude. Let \(\delta \mathbf{x} = \mathbf{x} - \bar{\mathbf{x}}\) (or equivalently \(\mathbf{x} = \bar{\mathbf{x}} + \delta \mathbf{x}\)). Substituting back and considering first-order approxmation yields to:

\[ f(\bar{\mathbf{x}} + \delta \mathbf{x}) \approx f(\bar{\mathbf{x}}) + \left. \frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} \right|_{\mathbf{x} = \bar{\mathbf{x}}} \delta \mathbf{x}. \]

https://stats.stackexchange.com/questions/5782/variance-of-a-function-of-one-random-variable