What is the difference between derivative and differential?

Roman Andronov, Senior Software Engineer at Dow Jones and Company (1999-present)

Answered Nov 29

Correct me if I’m wrong but I will take a small liberty of modifying the question slightly in order to make it mathematically meaningful:

what is the difference between a derivative of a function at a point and a differential of a function at a point?

The difference is infinitesimally small.

OK, that was a geeky joke. But seriously.

One way to investigate a difference between two objects is to examine their mathematically precise definitions - if those are available.

We are lucky with the definition of a derivative of a function at a point and not much so with the definition of a differential of a function at a point (see the discussion below).

We shall consider only real-to-real single variable functions.

Analytic Definitions

• Derivative of a function at a point

Let a function y(x)be defined on some interval I∈Rand let x0∈I. The derivative of y(x)at x0, denoted as y′x(x0), is, by definition, the following limit:

f′x(x0)=limx→x0y(x)−y(x0)x−x0(1)

regardless of whether the limit is a finite real number or is infinite.

If the limit in (1) is a finite real number then it is said that 1) y(x)is differentiable at x0 or 2) y(x)possesses/has a derivative at x0 or 3) a derivative of y(x) at x0 exists.

Small nitpick. Whether the function is in a single variable or in multiple variables, to avoid any potential ambiguities, it may be a good idea to make it painfully clear over which variable (symbol) the differentiation is carried out - differentiate function blah over (variable) blah.

The repetitive application of the limit in (1) to the resulting object on the left hand side of the equal sign in (1) produces the derivatives of higher order: the second derivative, the third derivative, … , the n$n$-th derivative of a function at a point.

We can granulate the definition of (1) further by distinguishing the direction (or the path) from which x0$x_0$ is approached - left or right.

I deduce from first principles (of (1)) the derivatives of a number of elementary functions in, these, Quora, answers.

• Differential of a function at a point

Let y(x)be differentiable at x0$x_0$. Then the differential of y(x)at x0, denoted dy or dy(x0), is by definition a function linear in dx:

dy=y′x(x0)dx(2)

where the dy symbol is atomic - it is not “dee times why”. As you develop the theory further and generalize, the differential will be a function in two variables, x and dx, dy=dy(x,dx).

A differential of a function can be used as a linear approximation of its actual change Δy in some neighbourhood of x0:

Δy(x0)=y(x0+Δx)−y(x0)

Example: let y(x)=x2, then:

Δy=(x0+Δx)2−x20=

x20+2x0Δx+Δx20−x20=

2x0Δx+Δx2

If x0=3and Δx=0.01 then:

Δy(3)=2⋅3⋅0.01+0.012=0.0601

While:

dy(3)=2⋅3⋅0.001=0.0600

We see that the difference, read - estimate/computational error, between the actual Δy and dy is just 0.0001.

The big idea here is that (we hope/expect that) as Δx becomes smaller and smaller (tends to zero) - so will the error. So if a function y(x) does posses a (finite) derivative at x0 then Δy can be represented as:

Δy=y′x(x0)dx+o(Δx)=dy+o(Δx)

where the little o function is usually interpreted as a computational error:

limΔx→0o(Δx)Δx=0

The concept of a differential scales well in both directions: to differentials of higher order and to differentials of functions in multiple variables.

Geometric Interpretation

Consider a graph g of y(x) and let P be a point on g. Construct a tangent τ to g through P (assume that it can be done).

Then the geometric interpretation of a derivative of y(x)at P is that it is the tangent (as a trigonometric function) of the angle θ that τ forms with the x-axis:

y′x(Px)=tanθ

Give Px a small (say) increase dxand this is where things get interesting.

Observe that our horizontal change dxis applicable in equal measure to both g and τ:

|PN|=dx

However, the corresponding vertical change is not:

The corresponding change for g is Δy:

Δy=|QN|

while the corresponding change for τ$\tau$ is dy$dy$:

dy=|TN|

One way to see that is as follows: we have one straight line τ falling on two straight parallel lines - the x-axis and PyP extended. Therefore, by Euclid’s B1.P29:

∠NPT=θ

Hence:

|NT|=|PN|⋅tanθ=y′x(Px)dx=dy

in accordance with (2).

Geometrically (or intuitively) speaking then, the differential of a function at a point is the change in the tangent straight line at a point caused by the change in the function’s independent variable. For a function in two variables its differential will be the change in the tangent plane and so on.

Discussion

In our chain of definitions we hung all our hats on the concept of a limit - a derivative (of a function at a point) was defined in terms of a limit and the corresponding differential was defined in terms of a derivative.

However, we meekly avoided saying anything about this mysterious magnitude of dx or Δx. Trouble is, in a course of a standard analysis dx can’t be defined rigorously by modern standards. All we can say about it is that it is “infinitesimally small” or “arbitrarily small”. If we all agree (and this is just an arbitrary agreement) that we can substitute y=x into our definition of the differential (2) then we might even “prove” that:

dy=dx=x′x⋅Δx=1⋅Δx=Δx

dx=Δx

and, hence, in a lot of textbooks the symbols dx$dx$ and Δx are used interchangeably (if these are used at all).

Historically, the way things shook out is that even though the ideas of the French mathematician Cauchy (1789–1857) (mostly) prevailed - the ϵ−δ-driven definition of a limit is the first class citizen in elementary analysis, the ideas and the notation due to Leibniz still linger on.

In M. Spivak’s “Calculus” on the bottom of the page 131 we find: “This ‘infinitely small’ quantity was denoted by dx … Although this point of view is impossible to reconcile with (P1)-(P13) properties of the real numbers, some people find this notion of the derivative congenial”.

In a Differential Geometry course you will be told that dx is a one-form, that it has to do with a point and a vector and a tangent space and so on.

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We’ve talked enough about tangent and derivative. To demonstrate that the fact that dxcan’t be rigorously defined in a standard analysis course is not the end of the world, let tangent and derivative make an encore appearance - let us manipulate the infinitesimally small entity to compute the derivative of a tangent (what else?)

To that end we steal a right triangle from the above drawing and relabel it to avoid confusion:

The triangle’s base is a unity, therefore, by definition:

tanθ=|BC||BA|=|BC|

Give θ an infinitesimally small increment dθ by rotating AC about Acounterclockwise (the drawing below is scaled up to gargantuan proportions for demonstration purposes):

Then, the small right (in the limit) triangle highlighted in red will be similar to △ABCby AAA (prove it). By Euclid’s B6.P4 (the sides about the equal angles in similar triangles are in the same proportion):

dtanθc⋅dθ=|AC||AB|=c1

dtanθdθ=c2

And by the Pythagorean theorem (or by B1.P47):

dtanθdθ=c2=1+tan2θ(3)

How much effort, would you say, was it to obtain (3)?

Infinitesimally small …