{
 "source": "https://peabody.kids/flashcards/math-derivative-rules",
 "id": "math-derivative-rules",
 "title": "Derivative Rules",
 "category": "math",
 "subject": "Math",
 "levels": [
  "9-12",
  "college"
 ],
 "description": "Differentiation rules and the derivatives of the common functions.",
 "tags": [
  "calculus",
  "derivatives"
 ],
 "frontLabel": "Function or rule",
 "backLabel": "Derivative",
 "cards": [
  [
   "Constant: d/dx [c]",
   "0"
  ],
  [
   "Power rule: d/dx [xⁿ]",
   "n xⁿ⁻¹"
  ],
  [
   "Constant multiple rule",
   "d/dx [c f(x)] = c f'(x)"
  ],
  [
   "Sum rule",
   "d/dx [f + g] = f' + g'"
  ],
  [
   "Difference rule",
   "d/dx [f − g] = f' − g'"
  ],
  [
   "Product rule",
   "d/dx [fg] = f'g + fg'"
  ],
  [
   "Quotient rule",
   "d/dx [f/g] = (f'g − fg') / g²"
  ],
  [
   "Chain rule",
   "d/dx [f(g(x))] = f'(g(x)) · g'(x)"
  ],
  [
   "d/dx [x]",
   "1"
  ],
  [
   "d/dx [x²]",
   "2x"
  ],
  [
   "d/dx [x³]",
   "3x²"
  ],
  [
   "d/dx [√x]",
   "1/(2√x)"
  ],
  [
   "d/dx [1/x]",
   "−1/x²"
  ],
  [
   "d/dx [eˣ]",
   "eˣ"
  ],
  [
   "d/dx [aˣ]",
   "aˣ 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"
  ],
  [
   "d/dx [cot x]",
   "−csc² x"
  ],
  [
   "d/dx [sec x]",
   "sec x tan x"
  ],
  [
   "d/dx [csc x]",
   "−csc x cot x"
  ],
  [
   "d/dx [arcsin x]",
   "1/√(1 − x²)"
  ],
  [
   "d/dx [arccos x]",
   "−1/√(1 − x²)"
  ],
  [
   "d/dx [arctan x]",
   "1/(1 + x²)"
  ],
  [
   "d/dx [sinh x]",
   "cosh x"
  ],
  [
   "d/dx [cosh x]",
   "sinh x"
  ],
  [
   "d/dx [e^(2x)]",
   "2e^(2x)"
  ],
  [
   "d/dx [sin(3x)]",
   "3 cos(3x)"
  ],
  [
   "d/dx [(x² + 1)³]",
   "6x(x² + 1)²"
  ],
  [
   "d/dx [x eˣ]",
   "eˣ(x + 1)"
  ],
  [
   "d/dx [x sin x]",
   "sin x + x cos x"
  ],
  [
   "d/dx [ln(x²)]",
   "2/x"
  ],
  [
   "d/dx [5x⁴ − 3x + 7]",
   "20x³ − 3"
  ],
  [
   "Derivative as slope",
   "The slope of the tangent line at a point"
  ],
  [
   "Limit definition of derivative",
   "f'(x) = lim (h→0) [f(x + h) − f(x)] / h"
  ],
  [
   "Second derivative",
   "The derivative of the derivative; tells concavity"
  ],
  [
   "Critical point",
   "A point where f' = 0 or f' does not exist"
  ],
  [
   "First derivative test",
   "f' changes from + to − at a local maximum, and − to + at a local minimum"
  ],
  [
   "Second derivative test",
   "f'' > 0 gives a local minimum; f'' < 0 gives a local maximum"
  ],
  [
   "Inflection point",
   "A point where concavity changes"
  ],
  [
   "Concave up",
   "f'' > 0; the graph curves like a cup"
  ],
  [
   "Concave down",
   "f'' < 0; the graph curves like a cap"
  ],
  [
   "Velocity from position",
   "v(t) = s'(t)"
  ],
  [
   "Acceleration from position",
   "a(t) = s''(t)"
  ],
  [
   "Mean value theorem",
   "A smooth curve has a point where the slope equals the average slope over the interval"
  ],
  [
   "L'Hôpital's rule",
   "For 0/0 or ∞/∞ forms, lim f/g = lim f'/g' (if that limit exists)"
  ],
  [
   "Implicit differentiation",
   "Differentiate both sides with respect to x, using dy/dx for y terms"
  ],
  [
   "Related rates",
   "Differentiate an equation in time to link rates of change"
  ],
  [
   "Tangent line equation",
   "y − f(a) = f'(a)(x − a)"
  ],
  [
   "Rolle's theorem",
   "If f(a) = f(b) and f is smooth, there is a point between where f' = 0"
  ]
 ]
}