Differential Calculus
What the exam tests
Questions test limits, continuity at a point, standard derivatives, product, quotient and chain rules, and the economic interpretation of derivatives as marginal cost, revenue or profit.
Core rules
- d/dx(xⁿ) = nxⁿ⁻¹; d/dx(eˣ) = eˣ; d/dx(log x) = 1/x; d/dx(constant) = 0.
- Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v².
- Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x).
- Marginal cost is MC = dC/dx; marginal revenue MR = dR/dx; profit is maximised where MR = MC and the second derivative is negative.
- A function is continuous at x = a if lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a).
Worked example
Differentiate y = x²eˣ. Let u = x², v = eˣ. Then u′ = 2x, v′ = eˣ. dy/dx = 2x·eˣ + x²·eˣ = eˣ(x² + 2x) = x eˣ(x + 2).
Business application: C(x) = 3x² + 5x + 100. MC = 6x + 5. At x = 10, MC = 65, meaning the approximate cost of producing the 11th unit is ₹65.
Traps
- Omitting the chain rule's inner derivative, e.g. writing d/dx(e^{3x}) = e^{3x}.
- Reversing the numerator in the quotient rule; the derivative of the denominator is subtracted.
- Treating a fixed cost as marginal cost - constants differentiate to zero.
- Using the derivative sign alone to claim a maximum without checking the second derivative.
- Confusing average cost C/x with marginal cost.