Math and natural sciences

How to learn university math on your own

People teach themselves university math for different reasons: to understand machine learning, to catch up before an exam, to read engineering or economics literature. The path is much the same for everyone — limits and derivatives, integrals, linear algebra, then differential equations and series. Below is the order, a rough timeline and ways to check that you really understand rather than just recognise formulas.

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Check your school foundation#

University math rests almost entirely on school math: manipulating expressions, exponents and logarithms, trigonometry, graphs of the basic functions. A quick test: without a textbook, sketch the graphs of y = x², y = 1/x, y = eˣ, y = ln x and y = sin x, and say where each one increases and where it is undefined. If that is hard, spend two or three weeks on functions and trigonometry — the precalculus part of learn math from scratch. It is not a step back: these are exactly the gaps people get stuck on in the very first chapter on limits. If the laws of exponents are not automatic, start there.

Part 1: limits and derivatives#

Start with intuition: a limit is the value a function approaches, and a derivative is its rate of change, the slope of the tangent line. First learn to find derivatives from the table and the rules (product, quotient, chain rule), then apply them: analyse a function, find its maximum and minimum, sketch its graph. The rigorous epsilon–delta definition of a limit can wait for a second pass — for applications, confident technique matters more.

Part 2: integrals#

The indefinite integral as the reverse of differentiation, the main methods — substitution and integration by parts — then the definite integral as an area and the fundamental theorem of calculus. Integration takes more practice than differentiation: there is no algorithm that covers every case, and the skill of recognising the right technique comes after a few dozen solved examples.

Part 3: linear algebra#

Vectors, matrices, determinants, systems of linear equations and Gaussian elimination, then linear transformations, eigenvalues and eigenvectors. If your goal is data analysis or machine learning, this part matters more than integrals. A geometric picture helps a lot: a matrix is a transformation of the plane, and the determinant tells you by what factor areas change. The idea of eliminating a variable is the same one you met in systems of linear equations, only scaled up.

Part 4: differential equations and series#

Separable first-order equations, second-order linear equations with constant coefficients, numerical and power series, expanding a function as a Taylor series. This part pulls together everything before it, so take it on once derivatives and integrals come without strain.

How long it will take#

At five or six hours a week, a rough guide: limits and derivatives take about two months, integrals one and a half to two, linear algebra two, differential equations and series another two. This is not a quota: at university the same topics are spread over three or four semesters, and if something goes slower, that is normal.

How to study so that you understand#

  • Alongside the textbook, watch visual explanations — calculus and linear algebra both have excellent free animated video series.
  • Rewrite every definition in your own words and invent an example and a counterexample for it.
  • Solve problems from a book with answers, and check derivatives and integrals in a computer algebra system such as WolframAlpha — but only after you have solved them yourself.
  • Once a week, go back to problems from earlier parts and solve them without hints — the active recall habit applied to math.

Step-by-step plan

  1. Weeks 1–3 — the school foundationFunctions and their graphs, logarithms, trigonometry; the sketching test on the basic functions.
  2. Months 1–2 — limits and derivativesIntuition for limits, the table of derivatives, the rules, analysing a function.
  3. Months 3–4 — integralsSubstitution, integration by parts, the definite integral and area.
  4. Months 5–6 — linear algebraMatrices, determinants, Gaussian elimination, eigenvectors with a geometric picture.
  5. Months 7–8 — equations and seriesFirst- and second-order differential equations, Taylor series.

Start learning this in your own space

The plan goes into your repository: tick off stages, keep notes — the change history shows how far you have come.

Start the plan

Check yourself

1.Find the value of the derivative of f(x) = x³ at x = 2.

2.Find the determinant of the 2×2 matrix with first row (2, 1) and second row (3, 4).

3.Evaluate the definite integral of 2x with respect to x from 0 to 1.

Sources

  • MIT OpenCourseWareFree MIT courses in calculus and linear algebra with lectures and problem sets
    free
  • 3Blue1BrownThe visual video series Essence of Calculus and Essence of Linear Algebra
    free
  • OpenStax Calculus Volume 1A free calculus textbook with worked examples and answers
    free

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