2.2 Defining the Derivative of a Function and Using Derivative Notation

The derivative as a limit

The derivative of f at a point x is defined as f'(x) = lim(h→0) [f(x+h) − f(x)] / h, provided the limit exists. This gives the instantaneous rate of change of f at x, and geometrically equals the slope of the tangent line to the curve at that point.

Notation matters

Several equivalent notations exist: f'(x), dy/dx, d/dx[f(x)], and y'. dy/dx emphasizes the derivative as a ratio of infinitesimal changes, while f'(x) emphasizes it as a new function. All represent the same rate-of-change concept, just written differently depending on context.

Differentiability requires continuity (not vice versa)

If a function is differentiable at a point, it must be continuous there — the converse is false. Functions fail to be differentiable at corners, cusps, vertical tangents, or discontinuities. Checking differentiability often means verifying the limit definition gives the same value from both sides.

Key terms

Limit definition of the derivative

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

Derivative notations

f'(x), dy/dx, d/dx[f(x)], and y' all represent the same derivative.

Geometric meaning of f'(a)

The slope of the tangent line to the graph of f at x = a.

Differentiability implies...

Continuity — but continuity does not imply differentiability.

Common non-differentiable points

Corners, cusps, vertical tangents, and discontinuities.

Try an AP-style question

lim(h→0) [(3 + h)² − 9] / h = ?

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