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Concept · Functions & Sequences
Applications of derivatives
monotonicity from sign of f', extrema/local max-min, closed-interval max/min, no-critical-point parameter ranges
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Key formulas
11- Definitionf'(x) = limh→0 [f(x+h) − f(x)]/h
- Tangent-line slope at x0k = f'(x0)tangent line y − f(x0) = f'(x0)(x − x0)
- Power ruled/dx xn = n xn−1d/dx (1/x) = −1/x2d/dx √x = 1/(2√x)
- d/dx ex = exd/dx ax = ax ln a
- d/dx ln x = 1/xd/dx loga x = 1/(x ln a)
- d/dx sin x = cos xd/dx cos x = −sin xd/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)
- Monotonicityf'(x) > 0 increasing, f'(x) < 0 decreasing
- Extremaf'(x0)=0 and f' changes + to − ⇒ local max− to + ⇒ local mincheck endpoints for closed-interval max/min
- Convexity (per CSCA convention)f''(x) < 0 on an interval defines a 'convex' function there
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsPrecalculus bridgeCalculus supportAlgebra 2 foundations