1. Let's start by stating the problem: We want to understand the differentiability of a function of two variables in a simple way. 2. Consider a function $f(x,y)$ that depends on two variables $x$ and $y$. Differentiability means we can approximate the change in $f$ near a point $(a,b)$ by a linear function. 3. The formula for differentiability at $(a,b)$ is: $$f(x,y) \approx f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b)$$ where $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to $x$ and $y$. 4. Partial derivatives $f_x(a,b)$ and $f_y(a,b)$ measure how $f$ changes when we change $x$ or $y$ slightly, keeping the other variable fixed. 5. If this linear approximation is very close to the actual change in $f$ when $(x,y)$ is near $(a,b)$, then $f$ is differentiable at $(a,b)$. 6. In simple terms, differentiability means the surface defined by $f(x,y)$ is smooth enough at $(a,b)$ so that it looks like a flat plane when zoomed in very close. 7. This is important because it allows us to use derivatives to analyze and predict how $f$ behaves near that point. Final answer: Differentiability of a function of two variables means the function can be well approximated by a plane near a point, using its partial derivatives.