Differentiability and differentials

In single-variable calculus, having a derivative at a point is the same as having a good tangent-line approximation there. In two or more variables, that equivalence quietly breaks: a function can have both partial derivatives $f_x$ and $f_y$ at a point yet still fail to be well-approximated by any plane through that point, because partials only look along the two coordinate directions and miss everything else. This article makes that gap precise with the definition of differentiability, gives the standard sufficient condition that most exam functions satisfy automatically, and introduces the total differential $df = f_x\,dx + f_y\,dy$ — the tool used to linearly approximate a function's value and to estimate how small measurement errors in the inputs propagate into a computed output.

The core insight

A function $f(x,y)$ is differentiable at a point when its change can be written as a linear part $f_x\,dx+f_y\,dy$ plus an error that vanishes faster than the increments, and this total differential then gives both a linear approximation of $f$ near that point and an estimate of error propagation.

The core idea

Differentiability is stronger than having partial derivatives. In one variable, $f$ differentiable at $a$ just means $f'(a)$ exists, and this automatically makes the tangent line a good local approximation. In two variables the analogous statement has to be built by hand: $f(x,y)$ is differentiable at $(a,b)$ if there exist constants $A,B$ such that

f(a+h,b+k)-f(a,b) = A\,h+B\,k+\varepsilon_1h+\varepsilon_2k,

where: $h,k$ are small increments in $x,y$ , $A,B$ turn out to equal $f_x(a,b),f_y(a,b)$ , and $\varepsilon_1,\varepsilon_2\to0$ as $(h,k)\to(0,0)$ . The mere existence of $f_x(a,b)$ and $f_y(a,b)$ does not force this — a function can behave well along the two axes alone yet badly along other directions, so it can have both partials at a point while failing to be even continuous there.

A usable sufficient condition. Checking the $\varepsilon_1,\varepsilon_2\to0$ condition directly is rarely convenient, so in practice one quotes: if $f_x$ and $f_y$ exist and are continuous throughout a neighborhood of $(a,b)$ , then $f$ is differentiable at $(a,b)$ . Every function built from elementary formulas (polynomials, sines, exponentials, and their combinations) satisfies this wherever it is defined, which is why differentiability rarely needs to be argued from the raw definition in routine problems.

The total differential. When $f$ is differentiable, the linear part $A\,h+B\,k$ is packaged as

df = f_x\,dx+f_y\,dy,

where: $dx,dy$ are small (possibly independent) increments in $x,y$ , and $f_x,f_y$ are evaluated at the base point. For $n$ variables this extends term by term, $df=f_{x_1}dx_1+\cdots+f_{x_n}dx_n$ , one contribution per variable with none dropped. Geometrically, $df$ is exactly the change predicted by the tangent plane to $z=f(x,y)$ at that point, while the true change $\Delta f = f(a+\Delta x,b+\Delta y)-f(a,b)$ is the actual surface height change; differentiability is precisely the statement that $\Delta f - df$ shrinks faster than the increments themselves, so $\Delta f\approx df$ for small increments.

Figure

A surface height change near a point compared with the tangent plane prediction, showing the true change and the linear approximation nearly coincide for a small step and diverge for a larger step

Derivation ATotal differential as the linear approximation of f

The goal is to show, starting only from the definition of differentiability at $(a,b)$ , that the total differential $df=f_x\,dx+f_y\,dy$ is exactly the correction term that turns $f(a,b)$ into the best available linear estimate of $f$ at a nearby point $(a+\Delta x,b+\Delta y)$ , with an error that becomes negligible for small increments.

Step 1. Start from the defining relation for differentiability at $(a,b)$ : there exist constants $A,B$ (equal to $f_x(a,b),f_y(a,b)$ ) such that $f(a+h,b+k)-f(a,b)=Ah+Bk+\varepsilon_1h+\varepsilon_2k$ , with $\varepsilon_1,\varepsilon_2\to0$ as $(h,k)\to(0,0)$ . Relabel the increments $h=\Delta x$ , $k=\Delta y$ to match how error-propagation problems are usually phrased.

Step 2. Substitute $A=f_x(a,b)$ and $B=f_y(a,b)$ into the relation. This gives $f(a+\Delta x,b+\Delta y)-f(a,b)=f_x(a,b)\Delta x+f_y(a,b)\Delta y+\varepsilon_1\Delta x+\varepsilon_2\Delta y$ , splitting the total change $\Delta f$ into a linear piece in $\Delta x,\Delta y$ plus a remainder built from $\varepsilon_1,\varepsilon_2$ .

Step 3. Define the total differential as exactly the linear piece, $df := f_x(a,b)\,dx+f_y(a,b)\,dy$ , writing $dx=\Delta x$ , $dy=\Delta y$ for the increments. Then $\Delta f = df + \varepsilon_1\Delta x+\varepsilon_2\Delta y$ , so the difference $\Delta f-df$ is exactly the remainder term that differentiability guarantees shrinks faster than the step size.

Step 4. Drop the vanishing remainder for small increments to get the usable approximation $\Delta f\approx df$ , i.e. $f(a+\Delta x,b+\Delta y)\approx f(a,b)+f_x(a,b)\Delta x+f_y(a,b)\Delta y$ . This is the working formula for both linear estimation of $f$ near a point and for estimating how a small error $dx,dy$ in the inputs propagates into an error $df$ in the output.

Worked example

A closed cylindrical can is measured with radius $r=5\text{ cm}$ and height $h=12\text{ cm}$ , and the volume is computed as $V=\pi r^2h$ . The measuring instrument has possible errors $dr=0.02\text{ cm}$ and $dh=0.05\text{ cm}$ . Estimate the resulting error in the computed volume, and express it as a percentage of the volume.

Step 1. Compute the partial derivatives of $V(r,h)=\pi r^2h$ : $V_r=2\pi rh$ and $V_h=\pi r^2$ .

Step 2. Evaluate the partials at the measured values $r=5,h=12$ : $V_r(5,12)=2\pi(5)(12)=120\pi$ and $V_h(5,12)=\pi(5)^2=25\pi$ .

Step 3. Form the total differential using the given error increments: $dV=V_r\,dr+V_h\,dh=120\pi(0.02)+25\pi(0.05)=2.4\pi+1.25\pi=3.65\pi\approx11.47\text{ cm}^3$ .

Step 4. Compute the actual volume and express the error as a percentage. $V(5,12)=\pi(25)(12)=300\pi\approx942.48\text{ cm}^3$ , so the relative error is $\dfrac{dV}{V}=\dfrac{3.65\pi}{300\pi}=\dfrac{3.65}{300}\approx0.01217$ , about $1.2\%$ . The computed volume is $942.48\text{ cm}^3$ with an estimated absolute error of about $\pm11.47\text{ cm}^3$ , roughly $1.2\%$ relative error.

Competitive stretch

The same setup can be flipped: given a maximum tolerable relative error in $V$ , find the maximum tolerable relative error in $r$ alone (holding $h$ exact). Since $V=\pi r^2h$ , writing $dV/V$ purely from the $r$ -term gives $\dfrac{V_r\,dr}{V}=\dfrac{2\pi rh\,dr}{\pi r^2h}=\dfrac{2\,dr}{r}$ — the relative error in $V$ from $r$ alone is twice the relative error in $r$ , a direct consequence of $r$ appearing squared in the formula. If the can's volume must be accurate to within $1\%$ using $r$ alone, $r$ itself must be measured to within $0.5\%$ , i.e. $dr\le0.005r=0.025\text{ cm}$ at $r=5$ .

Common traps

Assuming existence of $f_x,f_y$ automatically gives differentiability. False in general; the safe route is to quote the sufficient condition, continuity of $f_x,f_y$ near the point, rather than treating existence of the partials alone as enough.
Confusing $\Delta f$ with $df$ . They agree only approximately and only for small increments; for larger $\Delta x,\Delta y$ the differential can be a poor estimate because the neglected remainder is no longer negligible.
Dropping a term in the total differential. For an $n$ -variable function, every one of the $n$ terms $f_{x_i}\,dx_i$ belongs in $df$ — skipping a partial that looks small is a common source of error in multi-variable error-propagation problems.
Sign and unit errors in error-propagation problems. A decreasing measurement gives a negative increment (as with $\Delta y=-0.01$ in a shrinking-input example), and mixing absolute with relative error, or forgetting to convert units consistently, produces a wrong final estimate even when the calculus is right.

Ask the tutor: if the gap between "having partial derivatives" and "being differentiable" still feels abstract, ask the AI tutor to construct an example function with both partials at the origin that still fails to be differentiable there.

If a function has partial derivatives everywhere, why isn't that always enough for differentiability?

Partial derivatives only measure how $f$ changes along the two coordinate directions through a point — moving purely in $x$ or purely in $y$ . A function can be perfectly well-behaved along those two special directions while still oscillating badly or failing to have a consistent tangent plane along other directions through the same point. Differentiability demands good linear behavior in every direction at once, which is a genuinely stronger requirement. That is exactly why the sufficient condition — continuity of $f_x,f_y$ near the point — is so useful: it is what actually rules out this bad behavior for functions built from ordinary formulas.

When would df and the actual change Δf disagree noticeably?

They disagree once the increments $dx,dy$ are no longer small enough for the remainder term $\varepsilon_1dx+\varepsilon_2dy$ to be negligible compared to the linear part. In the volume example, the errors $dr=0.02$ and $dh=0.05$ are tiny relative to $r=5$ and $h=12$ , so $dV$ tracks the true change almost exactly. If instead $dr$ were, say, $2\text{ cm}$ , the quadratic dependence on $r$ in $V=\pi r^2h$ would make the linear estimate noticeably off from the true $\Delta V$ , because the neglected higher-order terms are no longer small.

How is the total differential different from just adding the two ordinary derivatives f prime with respect to x and y?

There is no single ordinary derivative for a two-variable function — "differentiating with respect to x and y separately" is exactly what the two partial derivatives $f_x,f_y$ already are. The total differential is not a new kind of derivative; it is a specific linear combination, $df=f_x\,dx+f_y\,dy$ , that accounts for simultaneous small changes in both variables at once. Treating it as some third derivative, rather than as the sum of the two partials' individual contributions weighted by their own increments, is the usual source of confusion here.

Differentiability and differentials

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads