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Any nonlinear differentiable function, i.e., a curve of points $$(y, x)$$, depicted as $$y = f(x)$$, can be locally approximated as a line, called linearization, around any point $$(\bar{y}, \bar{x})$$ on the curve.
One Independent Variable Case
A function of a single variable $$y = f(x)$$ can be approximately linearized around a point $$(\bar{y}, \bar{x})$$, where $$\bar{y} = f(\bar{x})$$, as
$$
y - \bar y \approx \frac{d f(\bar{x})}{d x} (x - \bar x_1).
$$
Two Independent Variable Case
A function of two variables $$y = f(x_1, x_2)$$ can be approximately linearized around a point $$(\bar{y}, \bar{x}_1, \bar{x}_2)$$, where $$\bar{y} = f(\bar{x}_1, \bar{x}_2)$$, as
A function of $$K$$ independent variables $$y = f( \mathbf{x})$$, where $$\mathbf{x} = (x_1, x_2, \ldots, x_K)$$, can be approximately linearized around point $$(\bar{y}, \bar{ \mathbf{x} })$$, where $$\bar{y} = f( \bar{ \mathbf{x} } )$$, as