A rectangle grown on two sides

Two quantities that change, multiplied together. Their product is the area of a rectangle with those two as its sides, and asking how the product changes is asking how the area of that rectangle grows when both sides are pushed out at once.

The picture is a dissection and it is checked as one: the four pieces are handed to the same assertion that the dissection proofs on this site use, which refuses a gap, an overlap, or anything outside the grown rectangle. What is being asserted is that the increase in area is exactly

$$u\,\Delta v + v\,\Delta u + \Delta u\,\Delta v,$$

which is an identity about numbers before it is anything about calculus.

The corner, and why it goes #

Divide that increment by the step $h$ and three terms appear. Two of them survive and one does not, and the difference between them is the entire content of the rule.

The strips are $v\,\Delta u$ and $u\,\Delta v$. Dividing by $h$ turns each into a length times a difference quotient, and difference quotients have limits. The corner is $\Delta u\,\Delta v$, which is a product of two things that each shrink like $h$; divided by $h$ it still shrinks like $h$, so it contributes nothing in the limit.

The right-hand column of the figure watches that happen. At the drawn step the corner is over sixteen per cent of the increment; six halvings later it is under one per cent, falling by about half each time, while the difference quotient of the whole product closes on $u'v + uv'$.

That is what second order means, and it is worth having a picture of, because the phrase is used constantly and is usually left as a formal remark. A second-order quantity is one built from two small factors, and it is invisible to a first derivative not because it is small but because it is small in a different way — small enough that dividing by one factor of $h$ still leaves it going to zero.

The rule, stated #

Taking the limit gives

$$(uv)' = u'v + uv',$$

and the picture makes both terms legible: each factor’s rate of change is weighted by the other factor, because the strip a growing side sweeps out has the length of the opposite side.

That weighting is the part worth remembering, and it is where most errors come from. The derivative of a product is not the product of the derivatives — the picture makes that obvious, since the product of the derivatives would be the corner alone, which is exactly the piece that vanishes.

The static ancestor #

The identity behind the picture is older than calculus and is usually met as an exercise in expanding brackets.

$(a+b)^2 = a^2 + 2ab + b^2$ is the case $u = v$, and the term that students routinely lose is $2ab$ — the two strips. In calculus the term that gets lost is the other one: the corner is the piece that should be dropped, and knowing which of the three pieces is negligible is the whole skill.

Setting $u = v$ in the rule gives $(u^2)' = 2uu'$, and repeating the argument with three factors gives $(uvw)' = u'vw + uv'w + uvw'$ — one term per factor, each with that factor differentiated and the others left alone. The picture for that is a box grown on three sides: three slabs, three edges and a corner cube, of which only the slabs survive.

The general case follows by induction and gives the power rule for whole exponents directly: $(u^n)' = n u^{n-1} u'$, with the $n$ counting the faces of an $n$-dimensional box. That is a satisfying way to see where the exponent in the power rule comes from — it is a count of faces, not an algebraic coincidence.

The discrete version, which is older #

Before any limit is taken, the identity is a statement about whole numbers, and it has been drawn for two and a half thousand years.

Going from $n^2$ to $(n+1)^2$ adds a gnomon of $2n + 1$ units: two strips of $n$ and one corner. As $n$ grows the corner becomes an ever smaller share of the addition — one part in $2n+1$ — which is the same observation made with counting rather than with limits.

The discrete identity is exact and needs no limit, and it is worth keeping in view for that reason. A difference equation for a product is

$$\Delta(u_n v_n) = u_{n+1}\Delta v_n + v_n \Delta u_n,$$

where one of the two factors has been shifted — a detail with no counterpart in the continuous rule, and the reason discrete calculus has two product rules rather than one. The shift is exactly the corner, allocated to one strip instead of being discarded, and there is no way to make the discrete identity symmetric without adding the corner back.

That asymmetry is a fair warning about the picture. The continuous rule looks symmetric in $u$ and $v$ because the corner is thrown away; before it is thrown away, a choice has to be made about which strip owns it.

Relative rates add #

Dividing the rule by the product itself gives a form that is often more useful than the rule:

$$\frac{(uv)'}{uv} = \frac{u'}{u} + \frac{v'}{v}.$$

The left-hand side is the relative rate of change of the product — the fractional change per unit — and the equation says that relative rates simply add. A quantity growing at three per cent multiplied by one growing at two per cent gives a product growing at five per cent, to first order, and the second-order correction is the corner.

That form is the logarithmic derivative, since each side is the derivative of a logarithm, and it is the cleanest statement of what logarithms are for: they convert multiplication into addition, so they convert the product rule into the statement that a sum’s derivative is the sum of the derivatives.

The reframing also explains why the corner is small in a way that needs no picture. Relative changes of a per cent or two produce a corner of a hundredth of a per cent, because the corner is the product of the two relative changes. The rule is exact to first order in the relative changes, and the error is second order in them — a statement with the same content as the drawing and none of its geometry.

The quotient, from the same rectangle #

A quotient is a product with one factor inverted, so the rule for it needs only the derivative of a reciprocal.

Take $w = 1/v$, so that $vw = 1$ and the rectangle has constant area. The increment identity then reads $v\,\Delta w + w\,\Delta v + \Delta v\,\Delta w = 0$: the two strips and the corner must cancel exactly, since the area does not change. Dividing by $h$ and taking the limit gives $vw' + wv' = 0$, so

$$\left(\frac{1}{v}\right)' = -\frac{v'}{v^2}.$$

The picture is a rectangle sliding along a hyperbola, thinning as it lengthens, and the minus sign is the visible fact that one side must shrink when the other grows. Combining that with the product rule gives the quotient rule, and it arrives with an explanation of its minus sign rather than as a formula to be memorised.

Written in relative terms it is even shorter: the relative rate of $1/v$ is minus the relative rate of $v$, and the relative rate of a quotient is the difference of the relative rates. That is the version worth carrying, and it is one line from the version above.

Read backwards, it is integration by parts #

The rule has a mirror image that is usually presented as a separate technique, and the picture makes them the same statement.

Integrate $(uv)' = u'v + uv'$ over an interval. The left side accumulates to the change in $uv$, by the fundamental theorem, so

$$\int u\,v'\,dx = \left[uv\right] - \int u'\,v\,dx.$$

In the language of the picture: the total area swept by the top strip equals the whole increase in the rectangle minus the total area swept by the side strip. Nothing has been introduced. The most useful technique of integration is one identity about a rectangle, integrated.

That is worth stressing because integration by parts is usually taught as a trick with a mnemonic, and its content is the same picture that gives the product rule. A reader who has the rectangle has both.

Where it needs care #

Three conditions are hiding in the drawing, and each is a real restriction.

Both factors must be differentiable. The corner argument needs both increments to be proportional to $h$; if one factor has a corner of its own, the strip belonging to it has no limit and the rule says nothing. A product of two non-differentiable functions can nevertheless be differentiable — $|x| \cdot |x| = x^2$ — so the rule’s failure is a failure to apply rather than a statement that the product misbehaves.

Order matters when the product does not commute. For matrices, the correct statement is $(AB)' = A'B + AB'$ with the factors kept in their original positions. The picture survives: the strips are still one factor times the other’s change, and the sides of the rectangle now carry an orientation. What fails is the temptation to write $BA'$ instead, and the fact that a product of matrices depends on the order is exactly what forbids it.

The strips must be strips. The argument assumes the increments are positive, so the four pieces genuinely tile the larger rectangle. If a factor decreases, the corresponding strip is subtracted rather than added and the picture becomes a signed sum. The identity survives unchanged — the algebra never used positivity — but the dissection does not, and the figure asserts both increments positive before it draws.

What it is for #

The rule is usually met as a step in a computation, and it is worth naming what it actually does, since the answer is not differentiating products.

It converts a question about a whole into a sum of questions about its parts. The rate at which a rectangle’s area grows is decomposed into a contribution from each side, weighted by how much of the other there is — and that decomposition is what makes it possible to reason about a compound quantity by reasoning about its ingredients separately.

Everything of the shape a total that is a product of factors inherits it. The rate of change of a volume is the sum of one term per dimension; the rate of change of a compound growth rate is the sum of its components’ rates; the sensitivity of a product to a small error in one factor is that factor’s error times the other factor. In each case the useful statement is not the formula but the decomposition: the effects of the factors on the whole are additive to first order, however multiplicatively they are combined.

That additivity is what makes calculus tractable at all, and it is the reason the corner has to be small. If the second-order term survived, the contributions of the two factors would interact and no decomposition of this kind would exist.

What the picture cannot show #

The corner shrinks and never vanishes. Every drawable step has a corner of positive area, and the claim is about the limit of a ratio, which no rectangle exhibits. The column of numbers is the honest half of the figure: it shows the share falling by half at each halving, and the rule is what that trend has as a limit.

The picture also handles one case of several. Both factors increase, both are positive, and both are functions of one variable. The rule holds without any of those, and for a function of several variables the corresponding statement is about a matrix of partial derivatives, whose picture is a different one.

Nor does the drawing establish the rule. It establishes an identity about four areas, which is exact and finite; passing from that to a statement about derivatives requires knowing that difference quotients converge, which is the content of the rung below.

Where it came from #

Leibniz worked out the rule in 1675 and got it wrong first. His notebook records the guess that the differential of a product is the product of the differentials — the corner alone — and then, ten days later, the correction. The published version, in 1684, is the modern statement.

The mistake is worth knowing about, because it is exactly the one the picture prevents. Anyone who has seen the rectangle cannot believe that the corner is the answer; it is visibly the smallest of the three pieces. Leibniz was reasoning symbolically, in a notation he was in the middle of inventing, and the notation had not yet acquired the geometric reading that makes the answer obvious.

Newton had the rule earlier, expressed in terms of what he called fluxions, and his derivation is essentially the rectangle argument with the corner discarded as a quantity too small to count — a step his critics attacked for a century and a half, and which is exactly the step the limit makes rigorous.

The ladder from here #

Below: the slope of a single point, which is the limit this rule is built from, and the curve with a corner everywhere, where that limit fails. Above: the flat map that fits closest, which is the same idea in several variables and where the corner term becomes a genuinely two-dimensional remainder, and the slope of a mirror image. Sideways: the fundamental theorem, which turns this rule into a method of integration.

The piece that is safe to throw away #

The lasting point is that the argument is a decision about which term to discard, and that the decision is principled rather than casual.

Every derivative is of this shape. An increment is written as a sum of pieces; the pieces proportional to the step survive division by it; the pieces proportional to its square do not. The whole of differential calculus is the systematic exploitation of that split, and the rectangle is the smallest picture in which the two kinds of piece can be seen side by side.

The same split, drawn differently, gives the derivative of a composition, the volume rule for a box, and — with the roles of the terms reversed — the error term of a tangent approximation, which is the corner promoted from a nuisance to the object of interest.