Lesson 20 of 48
The derivative
The slope of a curve at a single point.
A single point has no slope — until you take the limit of the slopes of shrinking secants. That trick is the entire engine of optimisation, and of the training loop at lesson 45.
Do this
Derive the derivative of x² from the limit definition by hand, showing every line. Then compute it numerically in Python for a small step h and watch the two converge.
The question that unlocks the next lesson
The derivative is defined as the limit of which quantity?
- Af(x + h) as h → 0
- B[f(x + h) − f(x)] / h as h → 0
- Cf(x) / x as x → 0
- Dthe area under f between x and x + h