2.2 Defining the Derivative & Notation

The derivative function

Letting the point vary gives the derivative function: f'(x) = lim(h→0) [f(x+h) − f(x)]/h. It outputs the slope of f at every x.

Notations

The derivative is written several ways: f'(x), dy/dx (Leibniz), d/dx[f(x)], and y'. Evaluating f'(x) at x = a gives f'(a), the slope at a.

Key terms

Derivative function

f'(x) gives the slope of f at every x.

Limit definition

f'(x) = lim(h→0) [f(x+h) − f(x)]/h.

Notations

f'(x), dy/dx, d/dx[f(x)], y'.

Leibniz notation

dy/dx — the derivative of y with respect to x.

Evaluating

f'(a) is f'(x) evaluated at x = a.

Try an AP-style question

lim(h→0) [(3 + h)² − 9] / h = ?

1. 6
2. 0
3. 3
4. 9