Derivatives from Graphs and Tables
- Estimate a derivative from tabulated values using a difference quotient
- Read the sign and relative size of f′ off the graph of f, and reconstruct f′ qualitatively
- Apply the product, quotient and chain rules to functions given only by a table of values
Most exam derivatives come without a formula
A large share of AP questions never give you f(x). They give a table of a few values, or a graph, and ask for f′ anyway. The rules still apply — the product rule does not care whether g′(2) came from differentiating a formula or from reading a table — but the evaluation has to be numeric from the start. Practice this separately, because students fluent in the rules routinely stall when there is nothing to differentiate symbolically.
The symmetric difference quotient
Given values at x = a − h and x = a + h, the best estimate of f′(a) is the symmetric difference quotient [f(a + h) − f(a − h)]/(2h). The denominator is the full width between the two x-values, not h. Halving that denominator is the single most common arithmetic error on table questions, and it doubles the answer. If the table only offers points on one side, a one-sided quotient is all you can do, and you should say the estimate is one-sided.
Reading f′ off the graph of f
Where f rises, f′ > 0; where f falls, f′ < 0; at a smooth peak or valley, f′ = 0. Steeper means larger |f′|. A straight segment of f means f′ is constant there, so the graph of f′ is a horizontal segment — not a line with the same slope. A corner in f means f′ has a jump discontinuity. Sketching f′ from f is really four questions asked at every x: rising or falling, how steeply, changing steepness, and is it smooth.
A table gives f(2) = 5, f′(2) = −3, g(2) = 4, g′(2) = 6. Let h(x) = f(x)·g(x) and k(x) = f(x)/g(x). Find h′(2) and k′(2).
- 1.Product rule: h′(x) = f′(x)g(x) + f(x)g′(x). No formula is needed — every piece is in the table.
- 2.h′(2) = (−3)(4) + (5)(6) = −12 + 30 = 18.
- 3.Quotient rule: k′(x) = [f′(x)g(x) − f(x)g′(x)] / [g(x)]².
- 4.Numerator at x = 2: (−3)(4) − (5)(6) = −12 − 30 = −42. Denominator: 4² = 16.
- 5.k′(2) = −42/16 = −21/8 = −2.625.
Before touching the arithmetic, write the rule with blanks: f′( )g( ) + f( )g′( ). Then fill the blanks from the table. Students who substitute while recalling the rule are the ones who end up with f′g′.
A table gives f(3) = 10 and f(7) = 22. Estimate f′(5).
The graph of f is a straight line segment with slope −2 on the interval (1, 4). What does the graph of f′ look like on that interval?
Table questions almost always ask you to justify the estimate. "f′(5) ≈ [f(7) − f(3)]/(7 − 3) = 3" earns the point; the bare number 3 often does not. Show the quotient you used.
Answer the 2 checkpoints as you read.
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