AP Calculus AB · Unit 2
Derivative Definition: every key term you need
20 flashcard terms for AP Calculus AB Unit 2, written to match the course framework. Study them here, then drill them as interactive flashcards — free, no account needed.
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Derivative Definition f'(a) = lim(h→0) [f(a+h)-f(a)]/h; instantaneous rate of change; slope of tangent line at point (a, f(a)).
Derivative as Limit Derivative exists when limit exists; geometrically, derivative is slope of tangent; physically, instantaneous velocity.
Tangent Line Equation At point (a, f(a)), tangent line: y - f(a) = f'(a)(x - a); uses point-slope form with derivative as slope.
Differentiability and Continuity If f differentiable at a, then f continuous at a. Converse false: continuous doesn't imply differentiable (sharp corners).
Power Rule d/dx[x^n] = n·x^(n-1); applies to any real n; fundamental rule used constantly in differentiation.
Constant and Constant Multiple Rule d/dx[c] = 0; d/dx[c·f(x)] = c·f'(x); constants differentiate to 0, constant factors pull out.
Sum and Difference Rule d/dx[f(x) ± g(x)] = f'(x) ± g'(x); differentiate term-by-term; linearity of derivative.
Product Rule d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x); 'first times derivative of second plus second times derivative of first.'
Quotient Rule d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)]/[g(x)]²; mnemonic: 'low d-high minus high d-low, square the bottom.'
Chain Rule d/dx[f(g(x))] = f'(g(x))·g'(x); derivative of composition; multiply outer and inner derivatives.
Derivative of Exponential d/dx[e^x] = e^x; d/dx[a^x] = a^x·ln(a); exponential functions have special property of self-reproduction.
Derivative of Logarithm d/dx[ln(x)] = 1/x; d/dx[log_a(x)] = 1/(x·ln(a)); inverse relationship with exponential.
Derivative of Trig Functions d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x), d/dx[tan(x)] = sec²(x); remember signs and patterns.
Inverse Trig Derivatives d/dx[arcsin(x)] = 1/√(1-x²), d/dx[arctan(x)] = 1/(1+x²); related to derivatives of original trig functions.
Implicit Differentiation Differentiate both sides with respect to x; treat y as function of x (use chain rule); solve for dy/dx; useful when y can't be isolated.
Related Rates Two variables related by equation; differentiate with respect to time; find rate of one variable given rate of another.
Logarithmic Differentiation Take ln of both sides, differentiate, solve for y'; useful for functions with variable exponents or complex products.
Higher Order Derivatives f''(x) = d/dx[f'(x)]; second derivative; f'''(x) is third derivative; notation: f^(n)(x) for nth derivative.
Concavity and Second Derivative f''(x) > 0: concave up (U-shaped); f''(x) < 0: concave down (∩-shaped); second derivative test for extrema.
Linear Approximation Near point a: f(x) ≈ f(a) + f'(a)(x-a); tangent line approximates curve locally; useful for estimating function values.
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