Implicit and parametric differentiation, quotient rule, and stationary points with integration and iteration

Last time

Before we start: last time, in Compound and double angles, R form and identities, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

One question, many methods

Paper 3 calculus questions chain several skills together. Differentiate when x and y are tangled, or when both are given in terms of a third letter. Then use the answer: set it to zero, integrate it, or iterate it. Each piece is routine. What earns the marks is knowing which piece to use first.

Implicit differentiation

When y is tangled with x, differentiate each term with respect to x. Every time you differentiate y, multiply by d y d x. For a product like x squared times y, use the product rule: 2 x y plus x squared d y d x. Then collect the d y d x terms, factorise, and divide. Substitute the point last.

A worked example

The curve is ln of x plus y, equals 3 x squared y. Differentiate the left: 1 plus d y d x, all over x plus y. The right: 6 x y plus 3 x squared d y d x. At 1, 0: the left is 1 plus d y d x over 1. The right is 0 plus 3 d y d x. So 1 plus d y d x equals 3 d y d x, and d y d x is one half.