Calculus I – Derivatives and Integrals
1. Limits
The limit of f(x) as x approaches a is L, written lim[x→a] f(x) = L, if f(x) can be made arbitrarily close to L
by taking x sufficiently close to a (but not equal to a). Key limit laws: sum, product, quotient, and
composition rules.
L'Hôpital's Rule
If lim f(x) = lim g(x) = 0 (or ±∞), then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.
2. Differentiation Rules
Rule Formula
Power Rule d/dx [x^n] = n·x^(n-1)
Product Rule d/dx [uv] = u'v + uv'
Quotient Rule d/dx [u/v] = (u'v - uv') / v²
Chain Rule d/dx [f(g(x))] = f'(g(x))·g'(x)
Exponential d/dx [e^x] = e^x
Natural Log d/dx [ln x] = 1/x
sin / cos d/dx[sin x]=cos x ; d/dx[cos x]=-sin x
3. Applications of Derivatives
3.1 Finding Critical Points
A critical point occurs where f'(x) = 0 or f'(x) is undefined. Use the First Derivative Test: if f' changes
from + to –, the point is a local max; if from – to +, a local min.
3.2 Concavity and Inflection Points
f is concave up where f''(x) > 0, concave down where f''(x) < 0. Inflection points occur where f'' changes
sign.
