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.

FoundationsPrecalculus bridge Calculus support Algebra 2 foundations

Applications of derivatives

monotonicity from sign of f', extrema/local max-min, closed-interval max/min, no-critical-point parameter ranges

No misconceptions are logged for this concept yet, so there is nothing named here to trip you up. Practice it and anything that does catch you will be added.

Practice this concept →