AP Calculus AB · Lesson 2 of 5
Differentiation
Compute derivatives with the standard rules and interpret the derivative as a rate or as the slope of a tangent line.
Rules and the chain rule
The derivative of a power xⁿ is n xⁿ⁻¹ for constant n. Sums and constant multiples pass through the derivative. A product needs the product rule, and a quotient needs the quotient rule. A composition needs the chain rule: the derivative of the outside, evaluated at the inside, times the derivative of the inside.
For f(x) = x³ − 2x at x = 2, f(2) = 4 and f′(x) = 3x² − 2, so f′(2) = 10. The tangent line is y − 4 = 10(x − 2). The derivative alone is the slope; the point still comes from the original function.
Meaning
If s(t) is position, s′(t) is velocity and s″(t) is acceleration. A positive velocity means the position is increasing, even if the position itself is negative. Units of the derivative are output units per input unit.
Differentiability implies continuity, but continuity does not imply differentiability. A corner can be continuous and still have no derivative.
Worked example
Find the equation of the tangent line to y = x² at x = 3.
- The point is (3, 9).
- y′ = 2x, so the slope at x = 3 is 6.
- The tangent line is y − 9 = 6(x − 3), or y = 6x − 9.
Why this works. Use the original function for the point and the derivative for the slope.
Check your understanding
- Identify products, quotients, and compositions before differentiating.
- Include the inner derivative when a function is composed.
- Interpret the sign of a derivative as increasing or decreasing, with units.