Type a function and a point x = a to get the first and second derivatives f′(a) and f″(a) as numbers.
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From the article Derivative at a Point — Meaning, Notation and Examples.
Question: Find f′(3) for f(x) = x², using the definition.
f(3 + h) = (3 + h)² = 9 + 6h + h²
f(3) = 9
f′(3) = limh→0 [ (9 + 6h + h²) − 9 ] / h
= limh→0 (6h + h²) / h
= limh→0 (6 + h) [cancel h, valid because h ≠ 0]
= 6
Check with the rule: f′(x) = 2x, so f′(3) = 2(3) = 6 ✔
Tangent line at x = 3: f(3) = 9, so
y − 9 = 6(x − 3) → y = 6x − 18 + 9 → y = 6x − 9
Worked in full in Derivative at a Point — Meaning, Notation and Examples.
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