Differentiability and Local Linearity 📈

Welcome, students! In this lesson, you will learn one of the most important ideas in multivariable calculus: how a function of two or more variables can be approximated by a flat surface near a point. This idea is called differentiability, and the flat approximation is called local linearity. These ideas matter because they let us estimate values, understand graphs, and predict how changing one input affects the output.

What you will learn

By the end of this lesson, you should be able to:

explain what it means for a multivariable function to be differentiable,
describe local linearity as a tangent plane approximation,
connect partial derivatives to the linear approximation of a function,
use the linearization to estimate nearby function values,
recognize why having partial derivatives alone is not enough to guarantee differentiability.

Think of this lesson as learning how to zoom in on a curved surface until it looks almost flat. That “almost flat” idea is the heart of local linearity ✨

The big idea: zooming in on a surface

In single-variable calculus, a differentiable function looks linear when you zoom in close enough to a point. For example, a smooth curve may look like its tangent line near one point.

In multivariable calculus, the same idea becomes more geometric. If $f(x,y)$ is a function of two variables, then its graph is a surface in three-dimensional space. Near a point $(a,b)$, a differentiable surface looks like a plane, not just a line. That plane is called the tangent plane.

This is what local linearity means: near $(a,b)$, the function can be approximated by a linear function of the changes in $x$ and $y$.

If $f$ is differentiable at $(a,b)$, then for small changes $\Delta x$ and $\Delta y$,

f(a+$\Delta$ x,b+$\Delta$ y) $\approx$ f(a,b)+f_x(a,b)\,$\Delta$ x+f_y(a,b)\,$\Delta$ y.

This is a powerful formula. It says the change in the output is approximately controlled by the partial derivatives. The partial derivatives measure the slopes in the $x$- and $y$-directions 🧭

Why this matters in real life

Local linearity is used whenever we need a quick estimate:

in engineering, to predict small changes in temperature, pressure, or stress,
in economics, to estimate how cost changes with production levels,
in science, to approximate how measurements vary when inputs shift slightly.

For example, if a formula gives the height of a surface or the output of a model, you often do not need the exact new value. A good approximation is enough, especially when the change is small.

Differentiability: more than just having partial derivatives

A common misunderstanding is that if a function has partial derivatives, then it must be differentiable. That is not always true.

The correct idea is stronger. A function $f(x,y)$ is differentiable at $(a,b)$ if it is well-approximated by a linear function near that point. In other words, the error between the actual function and its linear approximation becomes very small compared with the size of the input change.

Formally, $f$ is differentiable at $(a,b)$ if there exists a linear function $L(\Delta x,\Delta y)$ such that

$\lim_{(\Delta x,\Delta y)\to(0,0)}$

$\frac{f(a+\Delta x,b+\Delta y)-f(a,b)-L(\Delta x,\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$=0.

For a differentiable function, the best linear function is

$L(\Delta x,\Delta y)=f_x(a,b)\,\Delta x+f_y(a,b)\,\Delta y.$

That means the function is locally flat enough that its tangent plane gives an accurate approximation.

What the definition is saying in plain language

The formula above says the leftover error gets tiny faster than the distance you move away from the point. That is a very strong condition. It means the surface does not just have slopes in two directions; it behaves smoothly in all nearby directions.

So, students, differentiability is not only about measuring slopes. It is about whether those slopes fit together to make one good local linear model.

The tangent plane and the linearization formula

When a function $f(x,y)$ is differentiable at $(a,b)$, the tangent plane to the surface $z=f(x,y)$ at the point $(a,b,f(a,b))$ is

$ z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).$

This equation is also called the linearization of $f$ at $(a,b)$. It is the multivariable version of the tangent line.

The terms in the formula have clear meanings:

$f(a,b)$ is the height at the point,
$f_x(a,b)$ tells how steep the surface is when moving in the $x$ direction,
$f_y(a,b)$ tells how steep the surface is when moving in the $y$ direction,
$(x-a)$ and $(y-b)$ measure how far you moved from the point.

Example 1: estimating a nearby value

Suppose $f(x,y)=x^2+y^2$. Find the linearization at $(1,2)$.

First compute the partial derivatives:

$ f_x(x,y)=2x, \quad f_y(x,y)=2y.$

At $(1,2)$,

$ f(1,2)=1^2+2^2=5, \quad f_x(1,2)=2, \quad f_y(1,2)=4.$

So the linearization is

$L(x,y)=5+2(x-1)+4(y-2).$

If we want to estimate $f(1.02,1.98)$, use

$\Delta$ x=0.02$ and $$\Delta$ y=-0.02:

$ f(1.02,1.98) \approx 5+2(0.02)+4(-0.02)=4.96.$

The exact value is

$(1.02)^2+(1.98)^2=4.9608,$

which is very close. That is local linearity in action 🎯

How partial derivatives connect to differentiability

Partial derivatives are the building blocks of the linear approximation, but they are not the whole story.

If $f$ is differentiable at $(a,b)$, then both $f_x(a,b)$ and $f_y(a,b)$ must exist. However, the reverse is not always true. A function can have both partial derivatives at a point and still fail to be differentiable there.

A useful rule is this:

Differentiability implies partial derivatives exist at the point.
Partial derivatives alone do not guarantee differentiability.

A common sufficient condition is that if $f_x$ and $f_y$ are continuous near $(a,b)$, then $f$ is differentiable at $(a,b)$. This is a very important theorem in multivariable calculus.

Why continuity of partial derivatives helps

Continuity means the slopes do not suddenly jump. When the partial derivatives behave smoothly nearby, the surface is usually smooth enough to have a good tangent plane.

This is why many textbook problems ask you first to compute $f_x$ and $f_y$, then check whether they are continuous. If they are, differentiability is easy to conclude.

A nonexample: partial derivatives without differentiability

To see why extra care is needed, consider a function that behaves differently along different paths to a point. Such a function may have partial derivatives at the point but still fail to have a single good tangent plane.

For instance, some piecewise-defined functions have $f_x(0,0)$ and $f_y(0,0)$, but the limit needed for differentiability does not exist. This means the function cannot be well-approximated by one linear formula in every nearby direction.

The lesson is simple: to prove differentiability, do not stop after finding partial derivatives. You need to know that the error between the function and its linear approximation becomes negligible in every direction.

Interpreting local linearity geometrically

Imagine a smooth hill. If you stand on a tiny patch of the hill, the ground looks almost flat. The tangent plane is the mathematical version of that flat patch.

The graph of a differentiable function near a point behaves like this:

the surface curves overall,
but very close to the point, the curve is small enough that a plane matches it well,
the closer you zoom, the better the approximation becomes.

This is why linearization is so useful. Linear formulas are much easier to work with than curved surfaces, and they still give accurate local information.

In geometry terms, the tangent plane is the best plane approximation to the surface at the point. In algebra terms, the linearization uses the partial derivatives as coefficients. In reasoning terms, differentiability tells us that the local behavior of the function is predictable and smooth.

Conclusion

Differentiability and local linearity are central ideas in partial derivatives. They connect the slopes in the $x$ and $y$ directions to a full linear approximation of a multivariable function. If a function is differentiable at a point, then its graph has a tangent plane there, and the function can be estimated using its linearization.

The key message for students is this: partial derivatives tell you directional slope information, but differentiability tells you whether those slopes combine into a reliable local model. When that happens, the function behaves like a plane near the point, making approximation and analysis much easier ✅

Study Notes

Differentiability in multivariable calculus means a function is well-approximated by a linear function near a point.
Local linearity means the graph of $z=f(x,y)$ looks like a tangent plane very close to the point.
The linearization at $(a,b)$ is $z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$.
For small changes $\Delta x$ and $\Delta y$, $f(a+\Delta x,b+\Delta y) \approx f(a,b)+f_x(a,b)\Delta x+f_y(a,b)\Delta y$.
If $f$ is differentiable at a point, then $f_x$ and $f_y$ exist at that point.
The existence of partial derivatives does not by itself guarantee differentiability.
If $f_x$ and $f_y$ are continuous near a point, then $f$ is differentiable there.
Differentiability is the reason tangent planes and linear approximations work in multivariable calculus.
Local linearity is useful for estimation, modeling, and understanding small changes in real-world problems.
The main idea is to zoom in until a curved surface looks flat enough to treat like a plane.