Calculus 1B — The Derivative
Part 2 of 3. The derivative as a limit of slopes, tangent lines, the power/product/quotient rules, the chain rule, and implicit differentiation — with live Steps.
A derivative measures how fast something is changing at an exact instant — the steepness of a curve at a single point. It is the engine of all of differential calculus. This is part 2 of 3; it assumes you've met limits in part 1A.
- 01 The derivative as a limit of slopes ELI5: slope is "rise over run." Over a wide stretch of road that's just average steepness. But what's the steepness at one exact point?…
- 02 The tangent line ELI5: zoom far enough into any smooth curve and it looks straight — that straight line is the tangent, and its slope is the derivative. The tangent at x = a is…
- 03 Power, product & quotient rules ELI5: these are shortcuts so you never have to redo the shrinking-h limit by hand. Power + sum: differentiate each term with n·xⁿ⁻¹. Product: (f·g)′ = f′g +…
- 04 The chain rule ELI5: a function inside another function is like two meshed gears — turn the outer one and the inner one drives it. The chain rule says: multiply the outer…
- 05 Implicit differentiation ELI5: sometimes y is tangled up with x and you simply can't solve for it — like the circle x² + y² = 25. No problem: differentiate both sides as they are,…
next course: Calculus 1C — Applications of the Derivative →