C06 · Unit C2 Derivative foundations
Derivative definition and differentiability
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- Time
- About 45 minutes, untimed
- Course framework reference
- AP 2.1–2.4
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
The derivative f minutes (a) is lim as h tends to 0 of [f(a+h) minus f(a)]/h when a finite two-sided limit exists. It is the tangent slope and instantaneous output rate per input unit. Differentiability implies continuity, but corners, cusps and vertical tangents can obstruct a finite derivative. A secant gives an average rate; its limit, not one finite secant, gives the derivative.
Worked example
For f(x) equals x squared at a equals 3, the difference quotient is [(3+h) squared minus 9]/h equals 6+h for h is not equal to 0. Its limit is 6. Therefore the tangent is y minus 9 equals 6(x minus 3).
Explore the mathematics
Derivative definition and differentiability
Bring two points together and compare secant and tangent slopes.
Scroll sideways to see the full diagram.
Predict before continuing
Predict how the secant changes as the two points get closer.
Read the complete explanation
- Predict how the secant changes as the two points get closer.
- The orange secant joins two points: its slope is Δy/Δx.
- The green tangent has slope 2. Reduce h to see the secant approach it.
- The derivative is a limit, not substitution of h=0 into a quotient with zero denominator. Here f′(1)=2.
For y=x² at (3,9), the limiting secant slope is 6. The straight tangent y=9+6(x−3) matches that gradient at the marked point.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
The derivative limit is not evaluated by setting h equals 0 in the unsimplified quotient. At |x|, continuity at zero coexists with unequal one-sided slopes.
Investigate and explain
Compute secant slopes for x squared at x equals 3 with h equals 1,0.1, minus 0.1. Explain why the values approach 6 from different directions.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.