Units 2 and 3 cover the derivative — the instantaneous rate of change. This includes all differentiation rules and their applications to implicit equations and related rates problems.
Definition: f\'(x) = lim_{h→0} [f(x+h) - f(x)]/h. Power rule: d/dx(xⁿ) = nxⁿ⁻¹. Product: (uv)\' = u\'v + uv\'. Quotient: (u/v)\' = (u\'v - uv\')/v². Chain: d/dx[f(g(x))] = f\'(g(x))·g\'(x). Standard: d/dx(sinx) = cosx, d/dx(eˣ) = eˣ, d/dx(lnx) = 1/x, d/dx(aˣ) = aˣ ln a, d/dx(tan⁻¹x) = 1/(1+x²), d/dx(sin⁻¹x) = 1/√(1-x²).
Implicit differentiation: differentiate both sides with respect to x, applying chain rule to y terms (multiply by dy/dx). Then solve for dy/dx. Related rates: given how one quantity changes with time, find how a related quantity changes. Steps: (1) draw diagram, (2) write equation relating variables, (3) differentiate with respect to t, (4) substitute known values, (5) solve.
Tangent line at x = a: L(x) = f(a) + f\'(a)(x - a). Provides local linear approximation. If f\'\'(a) > 0 (concave up): linearisation underestimates. If f\'\'(a) < 0 (concave down): overestimates. This concept is tested frequently on both MC and FRQ.
The derivative f\'(a) gives the slope of the tangent line to the graph of f at the point (a, f(a)). Geometrically, it shows the instantaneous direction and steepness of the curve at that point. If f\'(a) > 0, the function is increasing (going uphill). If f\'(a) < 0, decreasing. If f\'(a) = 0, the tangent is horizontal (potential max/min). The derivative also represents the instantaneous rate of change — how fast the output changes relative to the input at that specific point.
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