Functions and Calculus
Core concepts
A Specialist candidate must move fluently between function notation, domain restrictions, and inverse/composite constructions. Know that a composite (f\circ g) exists only where (g(x)) lies inside the domain of (f), and that an inverse exists only on a restricted domain where the function is one-to-one. For calculus, distinguish differentiation from first principles (the limit of the difference quotient) from the shortcut rules, and be confident with the product, quotient, and chain rules, implicit differentiation, and parametric or inverse-trig derivatives. Integration includes substitution, recognising standard forms such as (\int \frac{f'(x)}{f(x)}dx = \ln|f(x)|+C), and definite integrals as signed areas.
Traps
Common errors: forgetting to change limits when substituting in a definite integral; dropping the (+C); assuming (f^{-1}(x)=1/f(x)); treating (\frac{dy}{dx}) as a fraction without justification; ignoring absolute values in logarithmic integrals; and misreading stationary-point questions that ask for nature via the second derivative or a sign table. Also watch domain-restricted square roots and denominators when stating ranges.
Worked example
Differentiate (y=\ln(\sin x)). By the chain rule, (\frac{dy}{dx}=\frac{\cos x}{\sin x}=\cot x). Rule of thumb: whenever a function sits inside a logarithm, differentiate the inside and divide by the inside.
Exam strategy
Show the rule you apply, state domains explicitly, and justify maxima/minima with a derivative sign argument, not just a sketch.