The Derivative as a Limit
- Interpret the derivative as the slope of the tangent line and an instantaneous rate of change
- Compute a derivative from the limit definition (difference quotient)
- Relate differentiability to continuity
From average rate to instantaneous rate
The slope of the secant line through two points on a curve, (a, f(a)) and (a + h, f(a + h)), is an average rate of change: [f(a + h) − f(a)]/h. As the second point slides toward the first — that is, as h → 0 — the secant pivots into the tangent line. Its slope is the instantaneous rate of change, and that limit is the derivative f′(a). The derivative turns "average speed over an interval" into "speed at an instant".
The limit definition
The derivative f′(x) is defined as the limit of the difference quotient, f′(x) = lim(h→0) [f(x + h) − f(x)]/h. Substituting h = 0 directly always gives 0/0, so every derivative from the definition is an indeterminate limit you must simplify — expand the numerator, cancel the h, and only then let h → 0. The rules you meet next are just this limit worked out once and for all for whole families of functions.
Differentiable implies continuous
If f is differentiable at a, then f is continuous at a — a function must be unbroken to have a tangent line. The converse fails: a continuous function can still fail to be differentiable, most famously at a corner (like |x| at x = 0), a cusp, or a vertical tangent, where the left and right slopes disagree or the slope is infinite. So "differentiable" is the stronger condition.
Use the limit definition to find f′(x) for f(x) = x².
- 1.Write the difference quotient: [f(x + h) − f(x)]/h = [(x + h)² − x²]/h.
- 2.Expand the numerator: (x² + 2xh + h²) − x² = 2xh + h².
- 3.Divide by h: (2xh + h²)/h = 2x + h (valid since h ≠ 0 inside the limit).
- 4.Take the limit as h → 0: 2x + 0 = 2x.
Continuity does not guarantee differentiability. The graph of y = |x| is continuous everywhere but has no derivative at x = 0, because the slope jumps from −1 to +1 at that corner. Always check for corners, cusps, and vertical tangents.
Using the limit definition, what is f′(x) for f(x) = 3x + 2?
A function f is continuous at x = 0 but its graph has a sharp corner there. What can you conclude?
When a problem says "using the definition of the derivative", you must show the limit of the difference quotient — the shortcut rules earn no credit on that specific prompt. Look for the h → 0 phrasing as your cue.
Answer the 2 checkpoints as you read.
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