Key Idea: This topic differentiates curves where y is not the subject, links rates that change over time, and finds the best value of a quantity. All three are favourite long questions on Paper 1 and Paper 2.
Implicit differentiation
- Differentiate — Every term with respect to x.
- y-terms — Each one leaves a dy/dx. xy-terms need the product rule.
- Rearrange — Collect the dy/dx terms and make dy/dx the subject.
- Use a point — Substitute it for the gradient, then y − y₁ = m(x − x₁).
Related rates of change
- Write the formula that links the quantities.
- Differentiate it with respect to time t.
- Substitute the known rate and value last.
- A negative rate means the quantity is shrinking.
Optimisation (HL contexts)
- Use the constraint to reduce to one variable.
- Set f′ = 0 and solve.
- Justify: f″ < 0 is a max, f″ > 0 is a min.
- Answer what is asked, and reject impossible values.
Horizontal tangent: top of dy/dx = 0. Vertical tangent: bottom = 0.
For x² + y² = 25, where is the tangent vertical? At (5, 0) and (−5, 0).
tan θ = h/100, h = 100, dh/dt = 5. Find dθ/dt. 1/40 = 0.025 radians per second.
Closest point on y = x² to (0, 3)? (±√(5/2), 5/2).
Why minimise D² instead of D? It gives the same x and is easier.