Finding the Linearization of a Function at a Point in 2D

Given the function $f(x, y) = 1 + x \cdot \ln(xy - 5)$, find the linearization of the function at the point $(2, 3)$.

Linearization is a powerful technique in calculus used to approximate the behavior of a more complex function near a specific point using a linear function. In multivariable calculus, this approximation involves using the tangent plane at the given point for the function of two variables. The concept hinges on understanding partial derivatives, as they are the key components in constructing the linear approximation, essentially capturing the function's rate of change in each variable direction.

To find the linearization of a function at a given point, we calculate the partial derivatives of the function with respect to each variable. These derivatives provide the slopes of the tangent plane in the respective directions of the variables. In essence, this method extrapolates the idea of a tangent line in single-variable calculus to a tangent plane in multiple dimensions. The linear approximation thus formed is particularly useful in estimating function values near the point of tangency, simplifying calculations and analyses around that region.

Comprehending the linearization process requires a solid grasp of multivariable calculus core concepts such as gradients and partial differentiations. Additionally, understanding how logarithmic functions transform inputs, particularly when involving products like xy, and how these dramatically affect the tangent plane's slope can further deepen one’s mathematical intuition.

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