HKDSE Maths M1 cheat sheet

Chain rule: never drop the inner derivative g'(x).

Integration by parts: choose u so du is simpler than u.

Change limits immediately when substituting in a definite integral.

Improper integrals require an explicit limit, not direct substitution.

Implicit differentiation collects dy/dx terms after differentiating both sides.

Median resists outliers; mean does not.

P(A ∪ B) subtracts the intersection to avoid double counting.

Conditional probability divides the joint probability by the condition's probability.

Combinations count selections; permutations count arrangements.

z = (x − μ)/σ places a value on the standard normal scale.

Random assignment, not random sampling alone, supports causal conclusions.

State domain restrictions before simplifying; cancelled factors still exclude values.

Composition f(g(x)) is evaluated inside-out and is generally not commutative.

Inverses require one-to-one functions; domain and range swap.

Horizontal transformations act opposite to the sign or factor shown.

Log laws convert sums to products, enabling solution of log equations.

Always check candidate solutions against original domain restrictions.

AP sum: n/2 × (first + last) or n/2(2a + (n − 1)d).

GP sum to infinity exists only for |r| < 1 and equals a/(1 − r).

The general binomial term C(n, r)a^(n−r)b^r gives any coefficient directly.

Count the number of terms carefully when the index starts at zero.

A negative common ratio alternates signs, so check the parity of the power.