Understand slopes, partial derivatives, the chain rule, and gradient descent through visual and coded experiments.
For a small example, minimize (w - 3)^2 starting at w = 0. Differentiate to get 2(w - 3), take a small step opposite the gradient, and watch the loss shrink.
A linear predictor has slope and intercept errors. Compute partial derivatives, update both parameters, and compare analytical gradients with finite differences.
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Predict how one control changes the Calculus & Gradients result.
The starter deliberately follows a tempting but flawed assumption: Reading a derivative as a global guarantee
Check: The browser checks pass and you can name the failure mechanism before opening the explanation.
Failing starter lab
Minimize a simple quadratic and compare its analytical gradient with a centered finite-difference estimate.
Run the incomplete starter before opening hints. Use the first wrong output or failed check to form a repair hypothesis.
Explanations after each answer · 80% to pass · Calculus & Gradients
Chapter 1 of 5 · Core
For a function f(x), the derivative f'(x) describes the local change in output per small change in input. It is the slope of the tangent line, not a promise about distant points. A positive derivative means a small increase in x tends to increase f locally; a negative derivative means it tends to decrease.
A finite difference approximates the derivative by comparing nearby function values. It is useful as a correctness oracle, although the step cannot be arbitrarily small because floating-point subtraction eventually becomes unstable.
Ideas to understand
Key equations
The Essence of Calculus
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Build a geometric intuition for derivatives and change before connecting gradients to model optimization.
Primary sources and free technical reading for Calculus & Gradients.
The official, free introduction to Python syntax and core programming ideas.
Free companion PDF covering linear algebra, calculus, probability, and optimization for ML.
Free course notes connecting gradients and optimization to neural-network training.