Calculus

Calculus answers two connected questions: how quickly is something changing, and how much does that change add up to? Start with $f(x)=x^2$, calculate with actual numbers, then name the general rule.

You need school algebra: substituting values, expanding brackets, and working with powers. Derivations and extra practice are tucked away when they interrupt the main explanation.

Rates of change

1. Derivatives: change at a point — Separate a function’s value, its rate of change, and a prediction for a nearby value.
2. Derivative rules, with the reasoning — See why powers, products, and nested functions have their particular rules.
3. Use slopes to find minima — Identify candidates, check whether they are minima, and take a downhill step.

Accumulation and several inputs

4. Integrals: add up a changing quantity — Build a total from small contributions, keeping track of units and signs.
5. From derivatives to integrals — Find an accumulated total by subtracting two antiderivative values.
6. Partial derivatives and gradients — Hold one input fixed, then predict what happens when several inputs change.

Use it

Practice starts with a compact lecture-style question and a translation of its notation. Work through new numbers in the mixed exercise, or choose a topic to revisit. The reference keeps the rules and their conditions together.

One dataset puts a gradient to work on a model’s actual prediction error. Backpropagation follows derivatives through a network.