Further Calculus Skills

calculus year12 ext1

Further calculus skills | NSW Curriculum Website

• ME1-12-04 selects and applies differentiation and integration techniques to solve problems

📖 Prior Knowledge

Content	Prerequisite relationships
Introduction to Differentiation	- Apply the chain rule → Differentiate parametric functions - Apply the chain rule → Derivative of an inverse function - Combine the differentiation rules → Differentiate combinations of functions - Combine the differentiation rules → Apply inverse trig differentiation
Differential Calculus	- Differentiate exponential functions → Differentiate combinations of functions - Differentiate ln x → Differentiate combinations of functions - Differentiate sin and cos → Differentiate combinations of functions - Differentiate combinations of functions (revisited content)
Integral Calculus	- Integrate power functions → Integrate to inverse trig forms - Apply the reverse chain rule → Integrate by substitution - Integrate trig functions of ax+b → Integrate ${\sin }^{2}nx$ and ${\cos }^{2}nx$
Further Work with Functions	- Find the inverse of a function → Derivative of an inverse function - Understand parametric equations → Differentiate parametric functions
Further Trigonometry	- Double angle identities → Integrate ${\sin }^{2}nx$ and ${\cos }^{2}nx$
Inverse Trigonometric Functions	- Define inverse trigonometric functions → Differentiate inverse trig 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 $(f^{-1}{)}^{\prime }(x)=\frac{1}{f^{\prime }(f^{-1}(x))}$
• 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 $\frac{d}{dx}[{\sin }^{-1}f(x)]=\frac{f^{\prime }(x)}{\sqrt{1-[f(x){]}^2}}$, $\frac{d}{dx}[{\cos }^{-1}f(x)]=-\frac{f^{\prime }(x)}{\sqrt{1-[f(x){]}^2}}=-\frac{d}{dx}[{\sin }^{-1}f(x)]$ and $\frac{d}{dx}[{\tan }^{-1}f(x)]=\frac{f^{\prime }(x)}{1+[f(x){]}^2}$ 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 $\int {\sin }^{2}nxdx$ and $\int {\cos }^{2}nxdx$