1.1 Derivatives

The derivative is the central object of calculus: a way to talk about rates of change with the same precision arithmetic uses to talk about quantities. Almost every physical formula on this bookshelf is a relation among derivatives, or an equation derived by setting a derivative equal to something. Newton’s laws, Maxwell’s equations, the wave equation, the heat equation — all are differential equations, which is to say all of them are statements about derivatives.

This first lesson is the working refresher. It introduces the derivative geometrically and algebraically, lists the four manipulation rules we lean on constantly, and ends with the priority dispute that shaped how we write derivatives today.

The derivative as a limit

The derivative of $f(t)$ at $t$ is defined as the limit

f'(t) \;=\; \frac{df}{dt} \;=\; \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}.

Geometrically it is the slope of the tangent line to the curve $y = f(t)$ at the point $t$ . Physically it is the instantaneous rate of change of $f$ — how fast $f$ would be changing if its current behaviour continued for an instant longer. When the independent variable is time we also write $\dot f$ ; when it is space, $f'$ . Both conventions appear across the books.

function:

x = 1.00

f(x) = 1.00 slope f′(x) = 2.00

The interactive shows the central definition of the derivative made visible: $f'(x)$ at a point is the slope of the tangent line. Pick a function, drag the point along the curve, watch the slope number track. If you take nothing else from single-variable calculus, take this picture — every other operation we do with derivatives (chain rule, optimisation, Taylor expansion, linearisation) is built on it.

⏳ The history — Newton, Leibniz, and why we have multiple notations

Differential calculus was developed independently by Isaac Newton in England (1665–1666, “fluxions”) and Gottfried Wilhelm Leibniz in Germany (1675–1684). The two formulations are mathematically equivalent but use different notation: Newton’s $\dot x$ for time derivatives, $\ddot x$ for second derivatives; Leibniz’s $df/dx$ , $d^2 f/dx^2$ . Leibniz’s notation generalises cleanly to multivariable calculus and made his approach dominant on the continent; Newton’s notation survived in physics and mechanics, where time is a privileged variable.

The dispute over priority — fuelled by national rivalries and by Newton’s accusations that Leibniz had plagiarised his work — soured Anglo-Continental mathematics for nearly a century. Britain stayed loyal to Newton’s clunkier “fluxional” calculus; the Continent ran with Leibniz’s notation and produced Euler, Lagrange, Laplace, and Fourier. The British eventually capitulated in the early 1800s. We use both notations today as a residue of the history: $\dot x$ for time, $\partial / \partial x$ for space, $f'$ when there is one variable and we don’t want to be fussy about which.

Read the original: Philosophiæ Naturalis Principia Mathematica (Isaac Newton (tr. Andrew Motte, 1729), 1687)

⏳ The history — Cauchy and the rigorisation of the calculus

For 150 years after Newton and Leibniz, calculus worked in practice but rested on shaky foundations. Newton’s “fluxions” and Leibniz’s “infinitesimals” were treated as quantities both vanishingly small and non-zero — a contradiction that Bishop Berkeley famously skewered in his 1734 pamphlet The Analyst, calling them “the ghosts of departed quantities.” Mathematicians used the methods because they worked; philosophers complained because they made no logical sense.

Augustin-Louis Cauchy’s 1821 Cours d’analyse and 1823 Résumé gave the modern definition of the derivative as a limit of difference quotients: $f'(x) = \lim_{h \to 0} [f(x+h) - f(x)]/h$ , with the limit defined by what we now call an $\varepsilon$ – $\delta$ statement. The reformulation eliminated infinitesimals entirely. Karl Weierstrass refined Cauchy’s definitions in the 1850s into the rigorous $\varepsilon$ – $\delta$ framework taught today.

This is the version of the derivative in the opening of this lesson — Cauchy’s, not Newton’s. The modern student inherits a calculus that has been logically clean for two centuries; the original was workable but informal for nearly as long as it has been rigorous.

The four manipulation rules

Most derivatives in this bookshelf are computed by combining four rules.

Linearity

(af + bg)' \;=\; a f' + b g'

for constants $a, b$ . The derivative of a sum is the sum of the derivatives, scaled by the constants. This is what allows Fourier methods: differentiating a sum of sinusoids gives a sum of (differentiated) sinusoids, with frequencies preserved.

Example: $\frac{d}{dt}\!\left[3\sin(t) + 2 t^2\right] = 3\cos(t) + 4 t$ .

Product rule

(fg)' \;=\; f' g + f g'.

The derivative of a product is not the product of the derivatives. Each factor in turn gets differentiated while the other holds still; sum the two contributions.

Example: $\frac{d}{dt}\!\left[t^2 \sin t\right] = 2t \sin t + t^2 \cos t$ .

The reason: writing $f g$ at $t + h$ and expanding, $(f + f' h)(g + g' h) = fg + (f'g + f g') h + f' g' h^2$ , and the $h^2$ term drops in the limit. Both factors contribute, but only their first-order pieces.

Chain rule

\bigl(f(g(t))\bigr)' \;=\; f'(g(t))\, g'(t).

The derivative of a composed function is the product of the outer derivative (evaluated at $g$ ) times the inner derivative. This is the single rule that most often “fails” when used carelessly — especially forgetting the inner $g'(t)$ factor.

Example: $\frac{d}{dt} \sin(t^2) = \cos(t^2) \cdot 2t$ .

composition:

x = 0.80

Drag x. The blue probe sits on g(x); its height u = g(x) sets the horizontal position of the red probe on f(u); the height of the red probe is h = f(u) = f(g(x)), which is the height of the black probe on the composed curve at the same x. Each panel's tangent line is the local rate of its own function. The chain rule below multiplies the two component rates to recover the slope of the composition.

The interactive shows the three pieces — inner $g(x)$ , outer $f(u)$ , and composition $h(x) = f(g(x))$ — for several common compositions. As you slide $x$ , watch how a change in $x$ first changes $g$ , which then changes $f$ . The product of those two rates is $h'(x)$ . The dashed gold guide lines trace the variable flow: $x \to u = g(x) \to h = f(u)$ .

Inverse rule

If $y = f(x)$ and $x = f^{-1}(y)$ , then

\frac{dx}{dy} \;=\; \frac{1}{dy/dx}.

The slope of an inverse function at a point is the reciprocal of the slope of the original function at the corresponding point.

Example: if $y = e^x$ , then $x = \ln y$ , and $\frac{d \ln y}{dy} = 1/e^x = 1/y$ . So $(\ln y)' = 1/y$ — the standard logarithmic derivative falls out of the inverse rule applied to the exponential.

Geometrically: the graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$ . Reflection swaps rise and run, so slopes are reciprocals.

Check yourself

Compute $\frac{d}{dt}\!\left[\cos(\omega t + \phi)\right]$ , with $\omega$ and $\phi$ constants. Which rule does this use, and why?

Reveal answer

$-\omega \sin(\omega t + \phi)$ . Chain rule: the outer function is $\cos(u)$ with derivative $-\sin(u)$ , evaluated at $u = \omega t + \phi$ ; the inner derivative is $\omega$ . The minus sign and the leading $\omega$ together are why every acoustic phasor calculation puts an $i\omega$ on time derivatives.

Check yourself

Given $y = \sin x$ , what is $\frac{dx}{dy}$ on the interval $(-\pi/2, \pi/2)$ , expressed in terms of $y$ ?

Reveal answer

By the inverse rule, $\frac{dx}{dy} = 1/\cos x$ . On the chosen interval $\cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - y^2}$ , so $\frac{dx}{dy} = 1/\sqrt{1 - y^2}$ — which is the standard derivative of $\arcsin y$ .

Check yourself

State the derivative as a limit, and use that definition to compute the derivative of $f(t) = t^2$ at an arbitrary point $t$ .

Reveal answer

$f'(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}$ . For $f(t) = t^2$ :

f'(t) = \lim_{h \to 0} \frac{(t+h)^2 - t^2}{h} = \lim_{h \to 0} \frac{2th + h^2}{h} = \lim_{h \to 0} (2t + h) = 2t.

The $h^2$ piece vanishes faster than $h$ and drops out in the limit — the same mechanism that lets the product rule keep only the cross terms.

Check yourself

Compute $\frac{d}{dt}\!\left[e^{-\gamma t} \cos(\omega t)\right]$ . Which two rules combine, and what’s the physical setting where this expression shows up?

Reveal answer

$-\gamma e^{-\gamma t}\cos(\omega t) - \omega e^{-\gamma t}\sin(\omega t) = -e^{-\gamma t}\!\left[\gamma \cos(\omega t) + \omega \sin(\omega t)\right]$ .

Product rule on the two factors, with the chain rule applied separately to each (derivative of $e^{-\gamma t}$ is $-\gamma e^{-\gamma t}$ ; derivative of $\cos(\omega t)$ is $-\omega \sin(\omega t)$ ). This is the velocity of a damped oscillator displaced as $x(t) = e^{-\gamma t}\cos(\omega t)$ — the central waveform of resonant systems.

Check yourself

Why does physics use $\dot x$ for time derivatives but $\partial / \partial x$ for spatial derivatives? Where does each notation come from?

Reveal answer

$\dot x$ is Newton’s “fluxion” notation for a derivative with respect to time — concise when time is the privileged variable (mechanics). $\partial / \partial x$ is Leibniz’s notation, generalised by Lagrange and others for partial derivatives in multivariable calculus. Both are equivalent; the choice is conventional. Mechanics texts lean Newtonian; field theory and analysis lean Leibnizian. Acoustics uses both: $\dot p$ for time derivatives of pressure, $\partial p/\partial x$ for spatial gradients in the wave equation.

Drill

Rote recall of the standard derivatives and the three combination rules, as a spaced-repetition deck. Reveal each card, then grade yourself — Again / Hard / Good / Easy — and SM-2 schedules when it returns. Progress is shared with the Foundations study deck.

What’s next

The next lesson, 1.2 — Integrals, develops integration: the inverse operation that recovers a function from its derivative. The rules for integrals mirror the rules for derivatives — substitution is the chain rule run backwards, integration by parts is the product rule run backwards — and together they form the closed loop that the fundamental theorem of calculus makes precise.