Differentiation
Differentiation questions test standard derivatives, the product, quotient and chain rules, and the second derivative. Commerce papers rarely demand first-principles proofs; they reward clean, rule-based differentiation with correct inner derivatives.
The derivative
dy/dx = lim(h→0)[f(x+h) − f(x)]/h measures the instantaneous rate of change. Standard results: d/dx(xⁿ) = nxⁿ⁻¹; d/dx(constant) = 0; d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ log a; d/dx(log x) = 1/x; d/dx(√x) = 1/(2√x); d/dx(1/x) = −1/x².
Rules
- Sum: (u ± v)′ = u′ ± v′.
- Product: (uv)′ = u′v + uv′.
- Quotient: (u/v)′ = (u′v − uv′)/v², v ≠ 0.
- Chain: d/dx f(g(x)) = f′(g(x)) · g′(x). The inner derivative is never optional.
- The second derivative d²y/dx² is the derivative of dy/dx and decides concavity.
Worked example
Differentiate y = (3x² + 1)(x − 2). Take u = 3x² + 1 and v = x − 2, so u′ = 6x and v′ = 1. Product rule: y′ = 6x(x − 2) + (3x² + 1)(1) = 6x² − 12x + 3x² + 1 = 9x² − 12x + 1. Expanding first gives 3x³ − 6x² + x − 2, whose derivative is the same, which confirms the result.
Traps
- Dropping the inner derivative, writing d/dx(3x + 5)⁴ = 4(3x + 5)³ instead of 12(3x + 5)³.
- Writing (uv)′ = u′v′.
- Differentiating a constant to 1 instead of 0.
- Reversing the quotient-rule numerator to uv′ − u′v, which flips the sign.