Differentiation and Integration Methods
Differentiation rules
This domain covers AS91578 (differentiation) and AS91579 (integration) and tests calculus as a problem-solving tool, not just routine differentiation. You need chain, product and quotient rules, derivatives of sin, cos, tan, e^x and ln x, implicit and parametric differentiation, rates of change, stationary points and optimisation. For integration, know standard results (including ∫1/x dx = ln|x| + c and ∫e^{kx} dx), substitution, definite integrals, areas between curves, volumes of revolution and simple separable differential equations.
Worked example
Differentiate y=x^2 ln x and find the tangent at x=1. Product rule: y'=2x ln x + x^2·(1/x)=2x ln x + x. At x=1, y=0 and y'=1, so tangent: y=x-1.
Integration
Reverse differentiation: ∫x^n dx = x^{n+1}/(n+1)+c for n≠-1. For substitution, change limits or substitute back. Area = ∫(upper-lower) dx; volume about x-axis = π∫y^2 dx. Differential equations: separate variables then integrate. A stationary point requires dy/dx=0, but you must justify whether it is a maximum, minimum or inflection using a sign table or second derivative. For volumes about the y-axis, use π∫x^2 dy with the correct limits.
Traps
Forget inner derivative in chain rule; quotient rule order (low d-high minus high d-low); omit +c; keep x-limits when using u; lose solutions by dividing by a variable; confuse dy/dx notation in implicit differentiation; forget to check endpoints in optimisation; sign errors in area/volume.