Short on time? This page is the recap. For explanations, the secant-line diagram and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (h → 0) g(h) means “the limit as h approaches 0 of g(h)”. f′(a) is read “f prime of a”.
Recap
- The average rate of change of f over an interval is change in output ÷ change in input: the slope of a secant line.
- The instantaneous rate of change at x = a is the limit of the average rate as the interval shrinks to a: the slope of the tangent line.
- That limit, when it exists, is the derivative of f at a, written f′(a). It is a number, with units of output per input.
- Both limit forms are 0/0 at first. Use the Topic 1.6 algebra (expand, factor, conjugate, combine fractions), cancel, then substitute.
- If the left and right limits of the quotient differ (for example at a corner), f′(a) does not exist.
Key relationships
| Idea | Formula | Notes |
|---|---|---|
| Average rate, h-form | (f(a + h) − f(a))/h | Interval between a and a + h, h ≠ 0 |
| Average rate, x-form | (f(x) − f(a))/(x − a) | Interval between a and x, x ≠ a |
| Link between forms | x = a + h, so x − a = h | Same number, different labels |
| Derivative at a, h-form | f′(a) = lim (h → 0) (f(a + h) − f(a))/h | Provided the limit exists |
| Derivative at a, x-form | f′(a) = lim (x → a) (f(x) − f(a))/(x − a) | Equivalent to the h-form |
Reading a limit backwards: find f from the f(a + h) or f(x) part, find a, and check that the number subtracted equals f(a). Example: lim (h → 0) ((3 + h)³ − 27)/h is f′(3) for f(x) = x³.
Assumptions behind the method
- h ≠ 0 (or x ≠ a) whenever you cancel. The limit never uses h = 0 itself.
- The limit must be a single finite value from both sides. Otherwise there is no derivative at a.
Mistakes to avoid
- Confusing f′(a) with f(a). One is a rate, the other a value.
- Writing f(a) + h for f(a + h). Substitute a + h for every x.
- Subtracting a or f(h) instead of f(a).
- Substituting h = 0 before simplifying. You always get 0/0.
- Stopping at the quotient. (f(a + h) − f(a))/h is still an average; take the limit.
- Leaving out units in a context question.
Quick self-check
- Find the average rate of change of q(x) = 5x − x² over [1, 4]. (0: q(1) = 4 and q(4) = 4. A zero average does not mean q never changed.)
- Use the definition to find q′(1). (3: the quotient simplifies to 3 − h for h ≠ 0)
- Evaluate lim (h → 0) ((3 + h)³ − 27)/h by recognising it. (It is f′(3) for f(x) = x³. For h ≠ 0 the quotient simplifies to 27 + 9h + h², so the limit is 27.)
Next: practice questions.