Differential Calculus
Differential calculus in the IPE papers runs from limits, continuity and differentiability through the standard derivative rules, implicit and parametric differentiation, logarithmic differentiation and second derivatives, into applications: tangents and normals, rates of change and maxima–minima of a function on an interval.
Core rules and concepts
- f is continuous at a if the left limit, right limit and f(a) agree; differentiability is stronger and implies continuity.
- Standard results to quote instantly: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(aˣ) = aˣ ln a, d/dx(ln x) = 1/x, d/dx(sin⁻¹x) = 1/√(1 − x²).
- For y = f(x)^g(x), take logarithms: ln y = g ln f, then differentiate implicitly.
- Parametric: dy/dx = (dy/dt)/(dx/dt); d²y/dx² = (d/dt of dy/dx)/(dx/dt).
- Tangent at (x₁, y₁): y − y₁ = (dy/dx)(x − x₁); the normal slope is the negative reciprocal.
- Stationary points from dy/dx = 0; classify by the sign change of dy/dx or by d²y/dx².
Worked example
Differentiate y = xˣ (x > 0). Take logarithms: ln y = x ln x. Differentiate both sides: (1/y)(dy/dx) = ln x + 1. So dy/dx = xˣ(1 + ln x).
Traps that cost marks
- Writing d/dx(xˣ) = x·xˣ⁻¹, which treats the exponent as a constant.
- Claiming a function is differentiable because it is continuous; |x| at 0 is the standard counterexample.
- Computing d²y/dx² for parametric curves as (d²y/dt²)/(d²x/dt²).
- Forgetting to check the endpoints when the question asks for absolute maximum or minimum on a closed interval.