LESSON 0.5 · Derivation · 100 min
Derivatives, partials, and the chain rule
Derivatives, partials, and the chain rule. Learn local slopes, Jacobians, computation graphs, and vector–Jacobian products and complete: Hand-calculate every local derivative in a
Learning objectives
- Explain what problem “Derivatives, partials, and the chain rule” solves without hiding behind terminology.
- Trace the variables and causal links across local slopes, Jacobians, computation graphs.
- Complete “Hand-calculate every local derivative in a two-layer function” and judge the result with evidence rather than intuition.
Core concepts
local slopes
local slopes is part of the lesson’s causal model. State its inputs, outputs, invariants, and failure mode; then verify it with a hand-check or a minimal experiment before moving to an optimized implementation.
Jacobians
Jacobians is part of the lesson’s causal model. State its inputs, outputs, invariants, and failure mode; then verify it with a hand-check or a minimal experiment.
computation graphs
computation graphs is part of the lesson’s causal model. State its inputs, outputs, invariants, and failure mode; then verify it with a hand-check or a minimal experiment.
and vector–Jacobian products
and vector–Jacobian products is part of the lesson’s causal model. State its inputs, outputs, invariants, and failure mode; then verify it with a hand-check or a minimal experiment.
Build and verify
Hand-calculate every local derivative in a two-layer function
- Predict: write the expected output, trend, or failure before running code.
- Build: implement only the minimum components needed to answer the question.
- Verify: compare with a baseline or trusted implementation; save seeds, parameters, and raw outputs.
- Transfer: change one shape, dataset, scale, or workload condition and explain whether the conclusion still holds.