2.1 Rates of Change at a Point

Derivative at a point

The derivative at a point is the limit of average rates as the interval shrinks: f'(a) = lim(h→0) [f(a+h) − f(a)]/h. It is the slope of the tangent line at x = a — the instantaneous rate of change.

Alternate form

An equivalent definition is f'(a) = lim(x→a) [f(x) − f(a)]/(x − a). The expression [f(a+h) − f(a)]/h is the difference quotient, which gives the average rate over [a, a+h] before the limit is taken.

Key terms

Derivative at a point

f'(a): the instantaneous rate of change / tangent slope at a.

Difference quotient

[f(a+h) − f(a)]/h; the average rate over [a, a+h].

Limit definition

f'(a) = lim(h→0) [f(a+h) − f(a)]/h.

Alternate form

f'(a) = lim(x→a) [f(x) − f(a)]/(x − a).

Tangent slope

The derivative at a is the slope of the tangent there.

Try an AP-style question

For f(x) = x² − 3x, what is lim(h→0) [f(2 + h) − f(2)] / h?

1. −2
2. 1
3. 0
4. 4