Course · Beginner
Calculus for Machine Learning
Limits, gradients and integrals: the mathematics of how models learn.
Calculus explains how gradient-based models learn. A loss measures how wrong the model is, its gradient says which way is downhill, and the chain rule carries that gradient back through every layer. This course builds calculus from limits to integrals with machine learning in view: derivatives of activation functions, the chain rule and vanishing gradients, gradients and contour maps, Jacobians and automatic differentiation, matrix calculus, Taylor series and the Hessian, convexity and Lagrange multipliers, and integrals from Riemann sums to neural ODEs. Each chamber runs the research loop, from discovery to derivation to real papers such as Robbins and Monro, Optimal Brain Damage and neural ODEs, with proofs to write and problems to code. The Chimera, a beast built by composition, waits at the centre.
- Chambers
- 9 + boss
- Time
- ~8 hours
- Questions
- 81
- Coding problems
- 27
- Concept cards
- 37
- XP available
- 4,970+
The path through the labyrinth
Three rings, 9 chambers, one guardian. You can wander ahead, but the thread works best in order: each chamber builds on the last.
The Outer Ring
Change in one dimension- 1Limits: Getting Arbitrarily CloseYou are hereWhat it means to approach without arriving: ε–δ, continuity, and the convergent series behind SGD's step sizes. 40 min 50 XP9 questions
- 2Derivatives: The Best Local LineThe derivative as the best linear approximation: prove the rules, then differentiate every activation in the zoo. 40 min 60 XP9 questions
- 3The Chain Rule: Deep CompositionsWhy derivatives multiply along a chain, proved properly, and why deep chains make gradients vanish or explode. 40 min 60 XP9 questions
The Middle Ring
Change in many dimensions- 4Gradients: Steepest Ascent in Many DimensionsPartial derivatives, directional derivatives and contour maps, and a proof that the gradient points straight uphill. 40 min 60 XP9 questions
- 5Jacobians and Automatic DifferentiationThe multivariable chain rule is matrix multiplication. Forward mode, reverse mode, and why one backward pass trains a network. 45 min 60 XP9 questions
- 6Matrix Calculus: Gradients of Layers and LossesDifferentiate with respect to vectors and matrices: least squares, linear layers and softmax cross-entropy, checked numerically. 45 min 60 XP9 questions
The Inner Ring
Curvature, constraints and accumulation- 7Taylor Series, the Hessian and CurvatureApproximate smooth functions with polynomials, read curvature from the Hessian, and tell minima from saddle points. 45 min 60 XP9 questions
- 8Optimisation: Convexity and ConstraintsWhen a local minimum is the global one, and how Lagrange multipliers solve problems that come with rules. 45 min 60 XP9 questions
- 9Integrals: From Riemann Sums to Neural ODEsArea as a limit of sums, the theorem that ties it to slope, and integrals at work in expectations, Monte Carlo and neural ODEs. 45 min 60 XP9 questions