Differential Calculus

calculus year11 advanced

Differential calculus | NSW Curriculum Website

• MAV-12-04 selects and applies differentiation methods to solve problems

📖 Prior Knowledge

Content	Prerequisite relationships
Trigonometry and Measure of Angles	- Graph trig functions in radians → Differentiate sin and cos
Trigonometric Identities and Equations	- Use the reciprocal identities → Differentiate tan, sec, cosec and cot
Introduction to Differentiation	- Differentiate sums and multiples → Differentiate sin and cos - Find tangents and normals → Find tangents and normals for any function - Apply the quotient rule → Differentiate tan, sec, cosec and cot - Apply the chain rule → Differentiate exponential functions - Apply the chain rule → Differentiate ln f(x) - Apply the chain rule → Differentiate composite trig functions - Combine the differentiation rules → Differentiate combinations of functions
Exponential and Logarithmic Functions	- Recognise $e^x$ is its own derivative → Differentiate exponential functions - Define the natural logarithm → Differentiate $a^x$ - Define the natural logarithm → Differentiate ln x

Differentiation with exponential functions

• Apply the rules of differentiation to find the derivative of $f(x)=k{e}^{ax}$, where k and a are constants
• Use the chain rule to prove $\frac{d}{dx}(e^{ax+b})=a{e}^{ax+b}$, where a and b are constants and $a\ne 0$
• Prove and use the formula $\frac{d}{dx}(a^x)=(\ln a)a^x$, where a is a constant and $a>0$
• Use the chain rule to differentiate functions of the form $e^{f(x)}$

Differentiation with logarithmic functions

• Use chain rule on the identity $e^{\ln x}=x$ to prove the formula $\frac{d}{dx}(\ln x)=\frac{1}{x}$ for the derivative of the natural logarithm function where $x>0$
• Prove and use the formula $\frac{d}{dx}({\log }_{a}x)=\frac{1}{x\ln a}$, where a is a constant and $a>0$
• Use the chain rule to differentiate functions of the form $\ln f(x)$

Differentiation with trigonometric functions

• Determine the formulas $\frac{d}{dx}(\sin x)=\cos x$ and $\frac{d}{dx}(\cos x)=-\sin x$ informally by using the graphs of $y=\sin x$ and $y=\cos x$ and verify with graphing applications
• Use the rules of differentiation to show that $\frac{d}{dx}(\tan x)={\sec }^{2}x$
• Use the rules of differentiation to find the derivatives of $\mathrm{cosec}x$, $\sec x$ and $\cot x$
• Use the chain rule to differentiate functions of the form $\sin f(x)$, $\cos f(x)$ and $\tan f(x)$

Using derivatives

• Apply the product, quotient and chain rules to differentiate functions of the form $f(x)g(x)$, $\frac{f(x)}{g(x)}$ or $f(g(x))$, where $f(x)$ and $g(x)$ are any of the functions within the scope of the Mathematics Advanced course
• Solve problems involving equations of tangents and normals to curves involving any of the functions within the scope of the Mathematics Advanced course