Chapter 2: Differentiation

2.1 Limit and Its Relation to Differentiation

• When x approaches a (x ≠ a) and f(x) approaches L, the limit is written lim_x→a f(x) = L. (p.31)
• If direct substitution gives 0/0 (indeterminate form), the limit cannot be found directly. (p.31)
• Methods to find a limit: direct substitution, factorisation, rationalising the numerator or denominator. (p.31)
• Rationalise by multiplying by the conjugate, using (a − b)(a + b) = a² − b². (p.32)
• First derivative by first principles: dy/dx = lim_δx→0 δy/δx = lim_δx→0 [f(x + δx) − f(x)]/δx. (p.35)
• Gradient of the curve at a point = gradient of the tangent = value of lim δy/δx. (p.35)
• dy/dx is a single symbol for the limit of δy/δx as δx → 0; it is not dy divided by dx. (p.35)
• Steps for first principles: take points A(x, y) and B(x + δx, y + δy); find δy = f(x + δx) − f(x); form δy/δx; take the limit as δx → 0. (p.36)

2.2 The First Derivative

• If y = axⁿ, then dy/dx = anxⁿ⁻¹ (a constant, n integer). (p.38)
• If n = 1, dy/dx = a; if n = 0, dy/dx = 0. (p.38)
• Gradient function f′(x) is a function; the gradient at a given point is a numeric value. (p.40)
• d/dx[f(x) ± g(x)] = d/dx[f(x)] ± d/dx[g(x)] (differentiate each term separately). (p.40)
• Chain rule: if y = g(u) and u = h(x), dy/dx = dy/du × du/dx. (p.42)
• For y = uⁿ: d/dx(uⁿ) = nuⁿ⁻¹ du/dx. (p.43)
• Product rule: d/dx(uv) = u dv/dx + v du/dx. (p.44)
• d/dx(uv) ≠ (du/dx)(dv/dx). (p.44)
• Quotient rule: d/dx(u/v) = (v du/dx − u dv/dx)/v², v ≠ 0. (p.45)

2.3 The Second Derivative

• The second derivative is d²y/dx² = d/dx(dy/dx), also written f″(x). (p.49)

2.4 Application of Differentiation

• Gradient of tangent f′(a) < 0: tangent slants to the left (negative gradient). (p.51)
• f′(a) = 0: tangent is horizontal (zero gradient). (p.51)
• f′(a) > 0: tangent slants to the right (positive gradient). (p.51)
• Line through P(x₁, y₁) with gradient m: y − y₁ = m(x − x₁). (p.53)
• Equation of tangent at P(a, f(a)): y − f(a) = f′(a)(x − a). (p.53)
• Equation of normal: y − f(a) = −[1/f′(a)](x − a), because m₁m₂ = −1. (p.53)

Turning Points and Their Nature

• A stationary point occurs when dy/dx = 0. (p.58)
• If dy/dx changes from positive to negative as x increases through a: maximum point. (p.58)
• If dy/dx changes from negative to positive: minimum point. (p.58)
• If dy/dx does not change sign: point of inflection (not a turning point). (p.58)
• A stationary point is a turning point only if it is a maximum or minimum point. (p.58)
• Second derivative test: dy/dx = 0 and d²y/dx² < 0 gives a maximum point. (p.60)
• dy/dx = 0 and d²y/dx² > 0 gives a minimum point. (p.61)
• If d²y/dx² = 0, use the tangent-sketching (sign of gradient) method. (p.62)

Maximum and Minimum Problems

• Method: express the quantity in one variable, solve dA/dr = 0, then use the second derivative to decide maximum or minimum. (p.63)
• Volume of a cylinder V = πr²h; use it to eliminate h and form the surface-area function. (p.64)

Rates of Change of Related Quantities

• If x and y change with time t and y = f(x): dy/dt = dy/dx × dx/dt (chain rule). (p.66)
• A negative rate of change means the quantity is decreasing. (p.67)
• For related-rate problems, use similar triangles to relate variables, e.g. r/5 = h/12 in the cone problem. (p.68)
• Sphere: V = (4/3)πr³, so dV/dr = 4πr². (p.69)

Small Changes and Approximations

• δy/δx ≈ dy/dx for small δx, so δy ≈ (dy/dx) × δx. (p.71)
• δy > 0: small increase in y; δy < 0: small decrease in y. (p.71)
• f(x + δx) ≈ y + (dy/dx)δx, i.e. f(x) + f′(x)δx. (p.71)
• Percentage change in x = (δx/x) × 100%; in y = (δy/y) × 100%. (p.72)
• Small-change problems use δy ≈ (dy/dx)δx with the relevant formula (e.g. V = 4/3 πr³). (p.74)