ConceptFunctions & Sequences

Key formulas

Definition: f'(x) = lim_h→0 [f(x+h) − f(x)]/h
Tangent-line slope at x0: k = f'(x0); tangent line y − f(x0) = f'(x0)(x − x0)
Power rule: d/dx xⁿ = n xⁿ⁻¹; d/dx (1/x) = −1/x²; d/dx √x = 1/(2√x)
d/dx e^x = e^x; d/dx a^x = a^x ln a
d/dx ln x = 1/x; d/dx log_a x = 1/(x ln a)
d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec² x
Rules: (u±v)'=u'±v'; (uv)'=u'v+uv'; (u/v)'=(u'v−uv')/v²; (cu)'=cu'
Chain rule: [f(g(x))]' = f'(g(x))·g'(x)
Monotonicity: f'(x) > 0 increasing, f'(x) < 0 decreasing
Extrema: f'(x0)=0 and f' changes + to − ⇒ local max; − to + ⇒ local min; check endpoints for closed-interval max/min
Convexity (per CSCA convention): f''(x) < 0 on an interval defines a 'convex' function there

If the concept underneath still feels loose, patch it here before returning to CSCA questions.

Derivatives, tangents, and applications

12 ways this goes wrong. Each one is a named misconception the questions are built to catch.