11–12Mathematics Extension 1 11–12 Syllabus

Implementation from 2026

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Content

Year 11

Year 12

Teaching advice

Examples

Further calculus skills

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME1-12-04
selects and applies differentiation and integration techniques to solve problems

Further calculus skills

Further derivatives of functions

Find the derivative of a function defined parametrically using the chain rule
Solve problems involving derivatives of functions defined parametrically
Verify using the chain rule that the derivative of the inverse function is the reciprocal of the derivative of the function, evaluated at the value of the inverse function, that is $(f^{- 1})' (x) = \frac{1}{f^{'} (f^{- 1} (x))}$
Solve problems involving derivatives of inverse functions
Examine the proofs of the derivatives of $\sin^{- 1} x$ , $\cos^{- 1} x$ and $\tan^{- 1} x$
Use the chain rule to show that $\frac{d}{dx} [\sin^{- 1} f (x)] = \frac{f^{'} (x)}{\sqrt{1 - {[f (x)]}^{2}}}$ , $\frac{d}{dx} [\cos^{- 1} f (x)] = - \frac{f^{'} (x)}{\sqrt{1 - {[f (x)]}^{2}}} = - \frac{d}{dx} [\sin^{- 1} f (x)]$ and $\frac{d}{dx} [\tan^{- 1} f (x)] = \frac{f^{'} (x)}{1 + {[f (x)]}^{2}}$ and apply the results to solve problems involving derivatives of $\sin^{- 1} f (x)$ , $\cos^{- 1} f (x)$ and $\tan^{- 1} f (x)$
Apply the product, quotient and chain rules to find derivatives of functions of the form $f (x) g (x),$ $\frac{f (x)}{g (x)}$ and $f (g (x))$ where $f (x)$ and $g (x)$ are any of the functions covered in the scope of the Mathematics Advanced 11–12 Syllabus (2024) or inverse trigonometric functions and solve related problems

Techniques of integration

Find indefinite and definite integrals involving expressions of the form $\frac{1}{\sqrt{a^{2} - x^{2}}}$ or $\frac{a}{a^{2} + x^{2}}$
Use integration by substitution to evaluate definite and indefinite integrals given the substitution, where the substitution is expressed as a function of the variable of integration or where the variable of integration is the subject of the substitution
Prove and use the identities $\sin^{2} nx = \frac{1}{2} (1 - \cos 2 nx)$ and $\cos^{2} nx = \frac{1}{2} (1 + \cos 2 nx)$ to find integrals involving $\int \sin^{2} nx dx$ and $\int \cos^{2} nx dx$

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