Q. 13 - Calculus

Question

What does it mean for a function of two variables, $f(x, y)$, to be differentiable at a point $(a, b)$? 

Step-by-Step Solution

Answer

$f(x,y)$ to be differentiated at some extent $(a,b)$ when it on an open set containing the purpose and if $\nabla f(x,y)=f(a+\Delta x,b+\Delta y)-f(a,b)$ be a function of $(x,y)$.

Step1: Differentiability.

A differentiable function of one real variable is $1$whose derivative occurs at each point in its domain, per mathematics.

In other words, each interior point within the domain of a differentiable function includes a non-vertical tangent line on its graph.

Step2: Differentiability for two functions of variables.

Let $f(x, y)$ be a function of two variables defined on an open set containing the purpose$(a, b)$, and  and let $\nabla f(x,y)=f(a+\Delta x,b+\Delta y)-f(a,b)$ be a function $(x,y)$.

If the partial derivatives$f_x(a, b)$ and $f_y(a, b)$ both exist, the function f is claimed to be differentiable at $(a, b)$.

$\nabla f(x,y)=f_x(a,b)\Delta x+f_y(a,b)\Delta y+{ϵ}_1\Delta x+{ϵ}_2\Delta y$

where ${ϵ}_1$ and ${ϵ}_2$ are $∆x$ and $∆y$ functions, and both move to zero when $(∆x,∆y)$$\to (0,0)$