Short on time? This page is the recap. For explanations, the tangent-line graph 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′(x) is read “f prime of x”.
Recap
- Keep the point as a variable x in the Topic 2.1 definition, and you get a function, f′.
- Inside the limit, x is fixed and only h changes. Cancel h (h ≠ 0) before letting h → 0.
- f′(x) exists only where the limit exists. It never exists where f is undefined, and it can fail where f is defined (√x at x = 0).
- f′(a) is a number: the slope of the tangent line at (a, f(a)), and the instantaneous rate of change at a.
- Units of f′ = units of f ÷ units of x.
- A derivative can be shown analytically (formula), numerically (table), graphically (tangent slopes) or verbally (a rate in words).
Key relationships
| Idea | Formula or notation | Notes |
|---|---|---|
| Derivative function | f′(x) = lim (h → 0) (f(x + h) − f(x))/h | Provided the limit exists |
| Notations for y = f(x) | f′(x), dy/dx, y′, d/dx (f(x)) | All mean the same thing |
| Value at a point | f′(a), or dy/dx at x = a | Find f′(x) first, then substitute |
| Tangent line at x = a | y − f(a) = f′(a)(x − a) | Point (a, f(a)), slope f′(a) |
| Results from the guide | d/dx (2x² − 3x + 1) = 4x − 3; d/dx (3/(x + 1)) = −3/(x + 1)² | Both found from the definition |
Mistakes to avoid
- Using the formula f′(x) as a tangent slope. The slope must be the number f′(a).
- Swapping height and slope. Height f(a), slope f′(a).
- Substituting first. f(2) is a constant; its “derivative” 0 is not f′(2).
- Writing f(x) + h for f(x + h).
- Splitting dy/dx into separate pieces in this unit.
- Missing units in a context answer.
Quick self-check
- Use the definition to find g′(x) for g(x) = x² + 4x. (2x + 4: the quotient is 2x + 4 + h for h ≠ 0)
- Find the tangent line to y = x² + 4x at x = 1. (Point (1, 5), slope 6: y − 5 = 6(x − 1), or y = 6x − 1)
- If P(t) is a population and t is in years, what are the units of dP/dt? (Individuals, such as people, per year)
Next: practice questions.