Differentiation and Its Applications
What the exam tests
Differentiation and Its Applications tests limits, derivative rules, tangent lines, monotonicity, and extrema. Kyoto entrance problems often require a sign chart and a clear distinction between local and global behavior on a stated interval.
Core rules
- The derivative at a is f'(a) = lim_{h→0} [f(a+h) - f(a)]/h, when the limit exists.
- Product, quotient, and chain rules: (fg)' = f'g + fg'; (f/g)' = (f'g - fg')/g^2; [f(g(x))]' = f'(g(x))g'(x).
- A tangent at x = a has slope f'(a) and equation y - f(a) = f'(a)(x - a).
- Critical points occur where f'(x) = 0 or f' is undefined. Use the sign of f' to decide increasing or decreasing; the second derivative can classify a stationary point when it is nonzero.
Worked example
For f(x) = x^3 - 3x^2 + 2, f'(x) = 3x^2 - 6x = 3x(x - 2). Critical points are x = 0 and x = 2. On (-∞, 0), f' > 0; on (0, 2), f' < 0; on (2, ∞), f' > 0. Thus f has a local maximum at x = 0 with f(0) = 2, and a local minimum at x = 2 with f(2) = -2. The tangent at x = 1 has slope f'(1) = -3 and passes through (1, 0), so its equation is y = -3(x - 1).
Traps
Do not call every critical point an extremum. Check endpoints when the interval is closed. Differentiate the inner function in a chain-rule problem. Distinguish the tangent line from the normal line, whose slope is the negative reciprocal.