Show that f(x, y)=x^2+3 y is differentiable at every point....

Show that $f(x, y)=x^{2}+3 y$ is differentiable at every point. In other words, show that $\Delta z=f(x+\Delta x, y+\Delta y)-f(x, y)=f_{x} \Delta x+f_{y} \Delta y+\varepsilon_{1} \Delta x+\varepsilon_{2} \Delta y,$ where both $\varepsilon_{1}$ and $\varepsilon_{2}$ approach zero as $(\Delta x, \Delta y)$ approaches $(0,0).$

Key Concepts

Error Terms and the Epsilon-Delta Definition

The error terms quantify the difference between the actual change in the function and its linear approximation. In the context of differentiability, these terms must become insignificant relative to the increment in the variables, formalized by the epsilon-delta criterion. This means the ratio of the error term to the change in the input should approach zero as the input change itself approaches zero.

Partial Derivatives

Partial derivatives measure the rate at which the function changes with respect to each variable individually, holding the other variables constant. They are the foundational components for computing both the gradient and the linear map that forms the linear approximation required for differentiability.

Linear Approximation (Tangent Plane Approximation)

Linear approximation involves using the first-order partial derivatives of a function to form a linear function that closely estimates the behavior of the original function near a specific point. This approximating plane, or tangent plane in two variables, provides a best-fit linear model for the function's local behavior.

Differentiability of Multivariable Functions

This concept refers to the idea that a multivariable function can be locally approximated by a linear transformation. The function is said to be differentiable at a point if there exists a linear map that approximates the change in the function for small perturbations in its inputs, with the error in the approximation becoming negligible relative to the size of the perturbation.

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