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Partial derivatives play a crucial role in understanding the behavior of multivariable functions like \( f(x, y) \). Unlike single-variable derivatives, partial derivatives focus on how a function changes with respect to one variable while keeping the others constant.
For example, if you have a function \( f(x, y) \), the partial derivative \( f_x(x, y) \) indicates how \( f \) changes as \( x \) changes, with \( y \) held constant. Similarly, \( f_y(x, y) \) measures the change in \( f \) when \( y \) changes.

• Notations: Common notations include \( \frac{\partial f}{\partial x} \) for the partial derivative with respect to \( x \).
• Calculation: Use similar rules to single-variable differentiation, applying them to one variable at a time.
• Applications: Partial derivatives are used in many fields like physics, engineering, and economics where relationships between variables are explored.

In our problem, understanding \( f_x(x_0, y_0) \) and \( f_y(x_0, y_0) \) is key, as they tell us how the function \( f \) behaves around the point \( (x_0, y_0) \). Knowing these helps us linearize \( g(x, y, z) \) and confirm its differentiability.

Linear approximation is a method used to estimate the value of a function near a given point using a linear function. It is particularly valuable when dealing with functions that are differentiable.
In essence, linear approximation treats a small segment of the curve of a function like a straight line. This line is determined by the function’s value and its derivatives at a particular point. For a function \( f(x, y) \), this approximation around the point \( (x_0, y_0) \) is given by:\[ f(x, y) \approx f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \] where the terms involving \( f_x \) and \( f_y \) account for how the function is changing in the \( x \) and \( y \) directions, respectively.

• Usefulness: It simplifies complex functions for easier analysis.
• Conditions: Requires the function to be differentiable at the point of interest.
• Applications: Widely used in economics to predict marginal changes, and in engineering to model small segment behaviors.

This concept directly applies to \( g(x, y, z) \) within our context since we aim to establish a similar kind of approximation indicating that \( g \) behaves linearly around the point \( (x_0, y_0, z_0) \).

Multivariable calculus expands the ideas of one-dimensional calculus into higher dimensions. Rather than focusing on single-variable functions, it examines functions of multiple variables. This expansion is crucial because many real-world problems involve more than one changing parameter.
In multivariable calculus, some key concepts include:

• Partial derivatives, which explore the rate of change of a function regarding one variable at a time.
• Gradient, which is a vector of all partial derivatives, indicating the direction of steepest ascent of a function.
• Double and triple integrals, which are used to calculate volumes under surfaces in higher dimensions.

The differentiability of a multivariable function, like \( g(x, y, z) \), depends on whether it can be linearly approximated in all directions around a point. Our task is rooted in these principles.
Differentiability here implies that you can create a flat plane (or hyperplane, in higher dimensions) that closely matches the function near \( (x_0, y_0, z_0) \). This understanding is crucial for working with real-world data in fields like physics, computer graphics, and optimization where predicting changes in multiple factors simultaneously is required.