What you'll learn
Applying derivatives to real contexts: rates, motion, and related rates.
Motion
For position s(t): velocity = s′(t), acceleration = s″(t). Speed = |velocity|. The object is speeding up when velocity and acceleration share a sign, slowing down when they differ.
Rates of change
The derivative is an instantaneous rate; interpret units in context (e.g. dV/dt in liters/min). Positive = increasing, negative = decreasing.
Related rates
When two quantities are linked by an equation and both change with time: differentiate the equation with respect to t (chain rule), then substitute known values.
- Steps: write the relationship, differentiate w.r.t. t, plug in the instant's values, solve for the unknown rate.
L'Hospital's rule
For limits of indeterminate form 0/0 or ∞/∞: lim f/g = lim f′/g′ (when conditions hold).
Exam tips
- Label what's given and what rate you're solving for.
- Keep variables as functions of t until after differentiating.
Common mistakes
- Substituting numbers before differentiating in related rates.
- Confusing speeding up/slowing down rules.