A derivative isn’t just a slope — it’s a rate of change, and rates run the real world. Track a particle’s motion, watch two quantities change together, ride a tangent line to approximate, and rescue a limit that collapses to 0/0. Five interactive tools that turn the derivative into a way of reading change as it happens.
Every derivative is a rate of change with units attached — gallons per minute, dollars per widget, degrees per second. Learn to say exactly what f′(a) means in a real scenario, read rates off graphs and tables, and never again confuse an amount with how fast it’s changing. This is the lens the whole unit looks through.
One particle, three linked functions: position, velocity, acceleration. Differentiate to descend the chain, read the signs to decide direction — and whether it’s speeding up or slowing down.
Two quantities tied by one equation, both changing in time. Differentiate the relationship with respect to t and their rates lock together — the ladder slides, the balloon inflates. Find the equation, differentiate, then substitute (never before).
Up close, every smooth curve looks like its tangent line. Ride that line — L(x)=f(a)+f′(a)(x−a) — to estimate. Concavity even tells you which way you missed: over a concave-up curve, the tangent always under-estimates.
When a limit collapses to 0/0 or ∞/∞, the answer hides in the rates. Swap the ratio of functions for the ratio of their derivatives and the fog clears — but first confirm the form is truly indeterminate. That one check keeps the rule honest.
Fresh batches of multiple choice across all five lessons, with worked explanations that name every distractor’s mistake and a weak-spot tracker — plus full free-response in the classic AP mold: the related-rates problem and the particle-motion problem, the two Unit 4 always brings to the exam.
Start practicing →Go in order. The derivative in context teaches you to read a rate’s meaning; Straight-line motion is that idea in the language of position, velocity, and acceleration; Related rates links two changing quantities; Linear approximation rides the tangent to estimate; and L’Hôpital’s rule rescues a limit that jams. Each tool animates the thing the textbook only describes.
Drill the two setups that lose points: the related-rates problem (name the equation, differentiate with respect to t, substitute last) and the motion sign-analysis (speeding up means v and a share a sign). The cheat sheet has every recipe, and the exam set mixes them the way the test will.