Find all points where the first-order partial derivatives of...

Find all points where the first-order partial derivatives of the functions are continuous. Then use Theorems 12.28 and 12.31 to determine the sets in which the functions are differentiable.
$$
f(x, y)=\frac{x}{x^{2}+y^{2}-1}
$$

Key Concepts

Differentiability in Multivariable Calculus

Differentiability of a multivariable function implies that the function can be well-approximated by a linear function near a given point. Theorem statements similar to Theorems 12.28 and 12.31 often assert that if the first-order partial derivatives are continuous in an open set, then the function is differentiable throughout that set. This connection between continuous partial derivatives and differentiability is fundamental in analyzing and approximating multivariable functions.

Continuity of Partial Derivatives

The continuity of first-order partial derivatives means that these derivatives do not have any abrupt changes or discontinuities in a neighborhood around a point. This property is important because, in multivariable calculus, if the partial derivatives of a function are continuous at a point (or in a neighborhood), then the function is typically differentiable at that point.

Domain of a Function

This concept refers to the set of all input values for which a function is defined. For functions involving quotients, the domain is typically restricted by conditions such as preventing division by zero. Analyzing a function’s domain is crucial because it identifies where subsequent operations like differentiation are valid.

First-Order Partial Derivatives

These are the derivatives of a multivariable function with respect to one variable at a time, treating the other variables as constant. They capture the instantaneous rate of change of the function in the directions of the coordinate axes and are fundamental in studying the local behavior of functions in higher dimensions.

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