#Calculus #Year12 #Ext1 >[!info]- [Further calculus skills | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-1-11-12-2024/content/n12/fa8f1cef28) >- ME1-12-04 selects and applies differentiation and integration techniques to solve problems ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ------------------------------------------------ | ----------------------------------------- | -------------------------------------------------- | | [[Area and Surface Area B]] | - area and surface area of various solids | - related rates of change | | [[Volume B]] | - volume of various solids | - related rates of change | | [[Differential Calculus]] | - derivative of exponentials | - exponential growth and decay | | [[Integral Calculus]] | - integration | - further integration, solids of revolution | | [[Stage 6 Extension 1/Polynomials\|Polynomials]] | - Polynomials | - Graphing polynomials using multiplicity of roots | | [[Inverse Trigonometric Functions]] | - inverse trigonometric functions | - derivatives of inverse trigonometric functions | ## 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 $\sin^2 nx=\frac{1}{2}(1-\cos 2nx)$ and $\cos^2 nx=\frac{1}{2}(1+\cos 2nx)$ to find integrals involving $\int \sin^2 nx\ dx$ and $\int \cos^2 nx\ dx$