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

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