Linearize a Multivariable Function at a Given Point

Linearize the multivariable function $f(x,y) = 1 + x \ln(xy - 5)$ at the point (2, 3).

Linearization is a powerful tool that helps in approximating complicated functions using simpler linear functions, especially in multivariable calculus. When we linearize a function at a point, we are essentially finding the tangent plane to the surface described by the function at that point. This process involves using partial derivatives to construct a linear function that closely approximates the original function near the intended point.

In this problem, you'll be working with a multivariable function that involves a natural logarithm, which adds an interesting twist to the process. The natural logarithm function, combined with a product of variables, demands careful attention to the behavior of the function not just at the point of interest, but also within its domain of definition to avoid invalid mathematical operations.

As you tackle this problem, remember to first compute the necessary partial derivatives of the function with respect to each variable. These derivatives provide the slope components for the tangent plane. Evaluating these derivatives at the given point and using the point’s coordinates allow you to construct the linear approximation—a tangent plane—in the multivariable context. By mastering this process, you build a foundational skill in approximating complex multivariable functions, which is particularly useful in understanding and predicting behavior locally in higher dimensions.

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