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Derivative Rules
Differentiation rules and the derivatives of the common functions.
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52 cards
- Constant: d/dx [c]Function or rule0
- Power rule: d/dx [xⁿ]Function or rulen xⁿ⁻¹
- Constant multiple ruleFunction or ruled/dx [c f(x)] = c f'(x)
- Sum ruleFunction or ruled/dx [f + g] = f' + g'
- Difference ruleFunction or ruled/dx [f − g] = f' − g'
- Product ruleFunction or ruled/dx [fg] = f'g + fg'
- Quotient ruleFunction or ruled/dx [f/g] = (f'g − fg') / g²
- Chain ruleFunction or ruled/dx [f(g(x))] = f'(g(x)) · g'(x)
- d/dx [x]Function or rule1
- d/dx [x²]Function or rule2x
- d/dx [x³]Function or rule3x²
- d/dx [√x]Function or rule1/(2√x)
- d/dx [1/x]Function or rule−1/x²
- d/dx [eˣ]Function or ruleeˣ
- d/dx [aˣ]Function or ruleaˣ ln a
- d/dx [ln x]Function or rule1/x
- d/dx [log_a x]Function or rule1/(x ln a)
- d/dx [sin x]Function or rulecos x
- d/dx [cos x]Function or rule−sin x
- d/dx [tan x]Function or rulesec² x
- d/dx [cot x]Function or rule−csc² x
- d/dx [sec x]Function or rulesec x tan x
- d/dx [csc x]Function or rule−csc x cot x
- d/dx [arcsin x]Function or rule1/√(1 − x²)
- d/dx [arccos x]Function or rule−1/√(1 − x²)
- d/dx [arctan x]Function or rule1/(1 + x²)
- d/dx [sinh x]Function or rulecosh x
- d/dx [cosh x]Function or rulesinh x
- d/dx [e^(2x)]Function or rule2e^(2x)
- d/dx [sin(3x)]Function or rule3 cos(3x)
- d/dx [(x² + 1)³]Function or rule6x(x² + 1)²
- d/dx [x eˣ]Function or ruleeˣ(x + 1)
- d/dx [x sin x]Function or rulesin x + x cos x
- d/dx [ln(x²)]Function or rule2/x
- d/dx [5x⁴ − 3x + 7]Function or rule20x³ − 3
- Derivative as slopeFunction or ruleThe slope of the tangent line at a point
- Limit definition of derivativeFunction or rulef'(x) = lim (h→0) [f(x + h) − f(x)] / h
- Second derivativeFunction or ruleThe derivative of the derivative; tells concavity
- Critical pointFunction or ruleA point where f' = 0 or f' does not exist
- First derivative testFunction or rulef' changes from + to − at a local maximum, and − to + at a local minimum
- Second derivative testFunction or rulef'' > 0 gives a local minimum; f'' < 0 gives a local maximum
- Inflection pointFunction or ruleA point where concavity changes
- Concave upFunction or rulef'' > 0; the graph curves like a cup
- Concave downFunction or rulef'' < 0; the graph curves like a cap
- Velocity from positionFunction or rulev(t) = s'(t)
- Acceleration from positionFunction or rulea(t) = s''(t)
- Mean value theoremFunction or ruleA smooth curve has a point where the slope equals the average slope over the interval
- L'Hôpital's ruleFunction or ruleFor 0/0 or ∞/∞ forms, lim f/g = lim f'/g' (if that limit exists)
- Implicit differentiationFunction or ruleDifferentiate both sides with respect to x, using dy/dx for y terms
- Related ratesFunction or ruleDifferentiate an equation in time to link rates of change
- Tangent line equationFunction or ruley − f(a) = f'(a)(x − a)
- Rolle's theoremFunction or ruleIf f(a) = f(b) and f is smooth, there is a point between where f' = 0
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