Find the increment Δz and write it in the form given in...

Find the increment $\Delta z$ and write it in the form given in (4.5). Determine whether $f$ is differentiable at all points $(a, b) .$ (Hint: Use a Taylor series for $e^{x}, \cos x,$ or $\sin x$.)
$$
f(x, y)=\frac{2}{x+y}
$$

Key Concepts

Increment of a Multivariable Function

This concept involves calculating the change in the value of a function when its inputs are perturbed. For a function f(x, y), the increment ?z is defined as f(x + ?x, y + ?y) - f(x, y). It is fundamental in analyzing how small changes in variables influence the overall function value and forms the basis for exploring more advanced ideas like differentiability and error estimations.

Differentiability of Multivariable Functions

A function of several variables is differentiable at a point if it can be closely approximated by a linear function near that point, with the error diminishing faster than the magnitude of the input change. This definition is formalized by expressing the function’s increment as a linear part (given by the partial derivatives) plus a remainder that becomes negligible, establishing the existence of a tangent plane at the point.

Taylor Series Expansion for Multivariable Functions

The Taylor series provides a method to approximate a function by a sum of its derivatives at a point, including terms of various orders. In the multivariable setting, the first-order Taylor expansion represents the linear approximation of the function, while higher-order terms account for the nonlinearity. This tool is essential for expressing the

Linear Approximation

Linear approximation, or the differential of a function, uses the first-order partial derivatives to construct the best linear estimate of the function near a specific point. It is the main component in expressing the function's increment in a form that separates the main linear behavior from the higher order error terms, thereby confirming differentiability.

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