Mathematics
Differentiability of Functions of Several Variables
Quick fact
A function of several variables can have all its partial derivatives at a point and still not be differentiable there. In fact, it can even have derivatives in every possible direction, yet fail to be differentiable—because differentiability requires a single linear approximation that works for all directions at once.
Why this is interesting
You've probably heard that a function is differentiable if you can find its partial derivatives. But in higher dimensions, that rule can completely mislead you—there are functions with partial derivatives at a point that aren't even continuous there. So what does differentiability really mean for a function of several variables?
Read the full explanation
Understanding Differentiability of Functions of Several Variables
Think of a single-variable function f(x) that is differentiable at a point a. There, you can draw a tangent line that closely approximates the graph near a. The slope of that line is the derivative, and the line is the best linear approximation. Now move to two variables: f(x, y). The graph is a surface. Differentiability at a point should mean that you can place a flat plane—the tangent plane—that approximates the surface well near that point. What does 'well' mean? In single-variable, it means that if you zoom in on the graph at a, the curve and the tangent line become indistinguishable. In several variables, it means that if you zoom in on the surface at the point, the surface and the tangent plane become indistinguishable. That's the intuitive idea. To make this precise, we define the total derivative (or just derivative) as a linear map. At a point p, a function F from R^n to R^m is differentiable if there exists a linear transformation L such that the error between F(p+h) and F(p) + L(h) is tiny compared to |h| as h approaches 0. In the special case where m=1 (a scalar-valued function), this L is a linear functional, often represented by the gradient vector. For m1, L is represented by the Jacobian matrix of partial derivatives. The catch: just having partial derivatives doesn't guarantee that such an L exists. Partial derivatives give you rates of change along specific axes, but differentiability demands that these rates combine coherently in every direction. A classic counterexample is a function that is zero along the axes but has a huge 'bump' along a curve that passes through the origin. Its partial derivatives at the origin might exist (both are zero), but the function isn't differentiable because as you approach the origin along that curve, the function doesn't approach the value predicted by the linear approximation.
A deeper explanation
The underlying principle is that differentiability is a local linear approximation condition. For a function F: R^n → R^m, the derivative at a point p is a linear transformation D F(p) that satisfies: lim{h→0} (F(p+h) - F(p) - DF(p)(h)) / |h| = 0. This means the error goes to zero faster than the length of h, so the linear map is a perfect tangent approximation. Why is this so important? Because a linear map is easy to work with—it respects addition and scalar multiplication. For instance, the chain rule in several variables says that the derivative of a composition is the composition of the derivatives. This is only true because we use the total derivative as a linear map; the chain rule is just matrix multiplication of Jacobians. Moreover, differentiability implies continuity: if a function is differentiable at a point, it must be continuous there. But the converse is false: continuous functions can fail to be differentiable. More surprising, even having all partial derivatives doesn't ensure differentiability. The partial derivatives only capture behavior along coordinate axes; differentiability requires that the linear approximation works regardless of the direction of approach. This is why directional derivatives are also not enough—they only look along straight lines, but the error condition must hold even along curved paths. Why does this matter beyond theory? In optimization, we use gradients to find minima. But without differentiability, the gradient may not point toward the steepest increase, and many methods fail. In physics, the laws of motion rely on velocity and acceleration being well-defined derivatives of position; if a trajectory weren't differentiable, concepts like instantaneous velocity would collapse. In geometry, the tangent plane is essential for defining curves and surfaces, and for calculus on manifolds. So differentiability is not just a technicality—it's the foundation that ensures we can locally approximate the world with linear models.