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Code Scholars Tutoring · AP Calculus AB

Derivatives as limits and local rates

Unit 2 · Topics 2.1, 2.2, 2.3. Original AP Calculus AB notes, worked example, and self-check.

Understand the idea

The derivative is the limit of secant slopes when that limit exists. It is both the slope of the tangent and an instantaneous rate with output-units per input-unit. Estimate from nearby table entries; a symmetric secant is often useful but still approximate. Distinguish a derivative value at one point from a derivative function on a domain.

f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h

Worked example

Use the definition to find the derivative of f(x)=x2\displaystyle f(x)=x^2 at x=3\displaystyle x=3.

The difference quotient is ((3+h)2−9)/h=(6h+h2)/h=6+h\displaystyle ((3+h)^2-9)/h=(6h+h^2)/h=6+h for h≠0\displaystyle h\ne0. Taking h→0\displaystyle h\to0 gives f′(3)=6\displaystyle f'(3)=6. The tangent slope at (3,9)\displaystyle (3,9) is 6\displaystyle 6.

Try it yourself

If f(1.9)=5.61\displaystyle f(1.9)=5.61 and f(2.1)=6.41\displaystyle f(2.1)=6.41, estimate f′(2)\displaystyle f'(2).

Show solution and reasoning

The symmetric secant estimate is (6.41−5.61)/(2.1−1.9)=4\displaystyle (6.41-5.61)/(2.1-1.9)=4. It is an approximation, not an exact consequence of two data values.

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