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Linearization is a technique used in multivariable calculus to approximate a function using its tangent line or plane at a particular point. This method simplifies complex functions into linear expressions that are easier to work with, especially for calculations or estimations close to the evaluated point.
For a function of several variables, like \( f(x, y, z) \), the linearization at a point \( (a, b, c) \) is expressed as:

• \( L(x, y, z) = f(a, b, c) + f_x(a, b, c)(x-a) + f_y(a, b, c)(y-b) + f_z(a, b, c)(z-c) \)

This formula includes the value of the function at the given point plus the contributions from each variable based on their partial derivatives. It's a way to grasp how changes in each variable affect the function in the immediate surrounding area of the point \( (a, b, c) \).
This concept is widely used in various scientific and engineering fields wherever multivariable functions appear. It is particularly useful when precise solutions are difficult to obtain, and approximations suffice.

Partial derivatives are fundamental in finding linearizations as they provide the rate of change of the function concerning each independent variable, while keeping the other variables constant. For the function \( f(x, y, z) = \tan^{-1}(xyz) \), its partial derivatives at a given point reveal how each variable individually influences the function's output.
Specifically:

• The partial derivative with respect to \( x \):\[ f_x = \frac{yz}{1+(xyz)^2} \]
• The partial derivative with respect to \( y \):\[ f_y = \frac{xz}{1+(xyz)^2} \]
• The partial derivative with respect to \( z \):\[ f_z = \frac{xy}{1+(xyz)^2} \]

These derivatives describe the slope of the tangent plane in the direction of the respective variable. The calculations show how sensitive the function is to changes in \( x \), \( y \), or \( z \) at any point.
Using partial derivatives is crucial in constructing the linear approximation since it helps determine the coefficients of each variable in the linear representation.

The tangent approximation, as a component of linearization, offers a simplified, linear form close to the original function when evaluated near the given point. Imagine drawing a tangent plane to the surface defined by the multivariable function; this plane acts as an approximate flat representation of the surface around the point of tangency.
To practically construct the tangent approximation, one calculates the function's value and its partial derivatives at a specific point, like \((1, 1, 1)\) for our function \( f(x, y, z) = \tan^{-1}(xyz) \). The linearization at this point results in the formula:

• \( L(x, y, z) = \frac{\pi}{4} + \frac{1}{2}(x-1) + \frac{1}{2}(y-1) + \frac{1}{2}(z-1) \)

This linear expression simplifies the complex behavior of the original function in a small neighborhood around the point. The tangent approximation becomes an excellent tool when you need to make quick calculations or predictions about the function's behavior without delving into complex computations.