←All concepts
Concept · Functions & Sequences
Derivatives, tangents, and applications
Practice this concept→
A Functions & Sequences set, untimed
Named traps
12 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Auxiliary function mis-signedBuilds the wrong auxiliary function (e.g. f(x)/ex) or mis-signs its derivative when comparing values.→
- Average vs instantaneous velocityConfuses average velocity over an interval with instantaneous velocity at a time.→
- Convex/concave f'' sign confusedConfuses convex with concave by using the wrong sign of the second derivative.→
- Derivative graph read as fInterprets the graph of f' as the graph of f itself.→
- Local min/max confusedPicks the local minimum critical point when the maximum is required (sign-chart direction).→
- No-critical-point discriminant strictnessUses discriminant<0 vs ≤0 incorrectly for 'no critical points' / convexity conditions.→
- Plugged into f instead of f'Evaluates the function instead of its derivative when finding a tangent slope.→
- Power-rule constant factor droppedForgets the factor (e.g. the 2 in d/dx x2) or the ln-base factor for ax.→
- Product-rule term missingDrops one term of the product rule (e.g. for x·ex or x·tan x).→
- Reports x not the pointReports the x-value of an extremum instead of the (x,y) point.→
- Tangent slope not negative reciprocalUses the line's slope instead of its negative reciprocal for a perpendicular-tangent condition.→
- f'(c) treated as variableTreats a constant f'(c) appearing in the definition as a variable when differentiating.→
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