Local analysis and asymptotic developments

Derivative at a point a ≠ 0

To find the derivative of f at a, we set h = x – a and find the derivative of g(h) = f(a + h) in the neighbourhood of h = 0, then return to x.

Example: Second-order derivative of ln(x) at a = 1. Let x = 1 + h: ln(1 + h) = h – h²/2 + o(h²), so ln(x) = (x – 1) – (x–1)²/2 + o((x–1)²)

Position relative to the tangent line

If f has a second-order derivative at a: f(x) = f(a) + f'(a)(x − a) + c*(x − a)^2 + o((x − a)^2), then y = f(a) + f'(a)(x − a) is the tangent at a, and the sign of c determines the position of the curve:

• c > 0: the curve lies above the tangent near a
• c < 0: the curve lies below
• c = 0: one must consider the next higher order (possible inflection point)

Asymptotic expansion towards +∞

To analyse an infinite branch, we set x = 1/t (as t approaches 0+) and find a local derivative of f(1/t) at 0, which gives an asymptote and the position of the curve relative to it.

Example: f(x) = √(x² + 1). As x approaches +∞: f(x) = x * √(1 + 1/x²) = x * (1 + 1/(2x²) + o(1/x²)) = x + 1/(2x) + o(1/x). The asymptote is y = x, and the curve lies above it because 1/(2x) > 0 for x > 0.

Common pitfall

Do not forget to factor out x when examining an asymptote at ±∞: the limit must be taken of an expression tending to 0 (in this case, 1/x), not of x itself.