Notation & identities to know
This is the lookup table for the volume. Every symbol the guide introduces, and every rule it proves, is collected here in one place — each row links back to the part that derives it, so a glance at a strange expression tells you both what it means and where to re-read the argument.
How to use this card
Filter, don't scroll
The two tables below cover the whole volume: the notation table reads a symbol and says where it comes from, and the identities table writes each rule in the form you will actually use it. Type into the box to filter every row as you go — it matches symbols, names and prose, case-insensitively. Clear it to see everything again.
No row matches that — try a looser word, or clear the box.
Notation
What the symbols mean, and where they were defined
| Symbol | Reads as | Derived in |
|---|---|---|
f(x), s(t) |
A function: an input mapped to an output. In the running example $s(t)$ is the robot's position at time $t$. | Part 1 · Zoom in until it's a line |
Δs / Δt |
An average rate of change over a finite interval — the slope of the secant line joining two points. | Part 3 · The derivative |
limh→0 |
The limit: the single value a quantity settles on as its variable approaches a target, if one exists. | Part 2 · Limits, made concrete |
(f(a+h) − f(a)) / h |
The difference quotient; the derivative is its limit as the run $h$ shrinks to zero. | Part 3 · The derivative |
f'(a), df/dx, ds/dt |
The derivative at a point, as a function of that point. Leibniz notation carries the units and behaves like a fraction under the chain rule. | Part 3 · The derivative |
f(a+h) ≈ f(a) + f'(a)h |
The best linear approximation: near $a$, the tangent line stands in for the curve with error of order $h^2$. | Part 1 · Zoom in until it's a line |
xn+1 = xn − f(xn)/f'(xn) |
Newton's method: follow the tangent line to its root, repeatedly. The same first-order model used as an algorithm. | Part 7 · Linear approximation, Newton, Taylor |
Tn(x), f(a+h) = Σ … |
The degree-$n$ Taylor polynomial of $f$ about $a$: the best polynomial stand-in that matches value and first $n$ derivatives. | Part 7 · Linear approximation, Newton, Taylor |
Σ, ∫ |
A finite sum and its continuum limit. The integral is signed accumulation — the Riemann sum taken to the limit of infinitely thin pieces. | Part 8 · The integral as accumulation |
A(x) = ∫ax f(t) dt |
Area-so-far: the accumulator whose rate of change, by the Fundamental Theorem, is the integrand $f(x)$. | Part 9 · The Fundamental Theorem |
F(b) − F(a) |
The net change in an antiderivative $F$ over $[a,b]$; it equals $\int_a^b f$ and turns area into subtraction. | Part 9 · The Fundamental Theorem |
O(h), O(h²) |
Big-O: the order of an error. $O(h)$ means halving the step halves the error; $O(h^2)$ means it quarters it. | Part 10 · Substitution, parts, and going numeric |
∂f/∂x |
A partial derivative: the slope in the $x$ direction with every other input held fixed. | Part 11 · Functions of several variables |
∇f |
The gradient: the vector of all partial derivatives, pointing in the direction of steepest ascent and normal to the level sets. | Part 12 · The gradient |
J, H |
The Jacobian (first derivatives, arranged as a matrix) and the Hessian (second derivatives, symmetric). | Part 13 · The derivative as a matrix |
fxxfyy − fxy2 |
The Hessian determinant used by the second-derivative test to sort each critical point into a minimum, a maximum or a saddle. | Part 14 · Multivariable Taylor & critical points |
Identities & rules
One line each, in the form you use them
| Rule | Statement | Derived in |
|---|---|---|
| Power rule | $\dfrac{d}{dx}\,x^{n} = n\,x^{\,n-1}$ | Part 4 · A toolbox of derivatives |
| Product rule | $(fg)' = f'g + fg'$ | Parts 4–5 |
| Quotient rule | $\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^{2}}$ | Part 4 · A toolbox of derivatives |
| Chain rule | $(f \circ g)'(x) = f'\bigl(g(x)\bigr)\,g'(x)$ | Part 5 · The chain rule |
| Implicit differentiation | $\dfrac{dy}{dx} = -\,\dfrac{F_x}{F_y}$ | Part 6 · Implicit differentiation & related rates |
| FTC — differentiate the accumulator | $\dfrac{d}{dx}\int_{a}^{x} f(t)\,dt = f(x)$ | Part 9 · The Fundamental Theorem |
| FTC — evaluate an antiderivative | $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$ | Part 9 · The Fundamental Theorem |
| Substitution | $\int f\bigl(g(x)\bigr)\,g'(x)\,dx = \int f(u)\,du$ | Part 10 · Substitution, parts, and going numeric |
| Integration by parts | $\int u\,dv = uv - \int v\,du$ | Part 10 · Substitution, parts, and going numeric |
| Taylor with remainder | $f(a+h) = \displaystyle\sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}\,h^{k} + R_n(h)$ | Part 7 · Linear approximation, Newton, Taylor |
| Second-derivative test | At a point with $\nabla f = 0$: $H$ positive definite $\Rightarrow$ local minimum, negative definite $\Rightarrow$ local maximum, indefinite $\Rightarrow$ saddle. | Part 14 · Multivariable Taylor & critical points |
Where to go next
The machinery this card points at
Everything above is single-variable calculus plus a first look at partial derivatives. Calculus in Motion picks up exactly where the matrix notation leaves off: the two-link arm returns, the Jacobian becomes a velocity map, and the same $J$, $H$ and $\nabla f$ from Parts 11–13 are put to work on a moving robot. When those derivatives are chained together and fed backwards through a network, the result is backpropagation — the derivative as an algorithm.
For the destination, Nonlinear Optimization is the natural continuation: it treats the second-derivative test and the gradient as the ingredients of an iterative solver, and asks what happens when the landscape is too large to write down. Keep this page open beside it — every step an optimizer takes is one of the rules in the table above.