#Calculus #Year11 #Advanced >[!info]- [Differential calculus | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n12/fa4d003480) >- MAV-12-04 selects and applies differentiation methods to solve problems ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ------------------------------------------ | --------------------------------------- | ----------------------------------------------------- | | [[Trigonometric Identities and Equations]] | - inverse trig functions | - derivative of inverse trig functions | | [[Introduction to Differentiation]] | - differentiation | - finding and interpreting the derivative | | [[Exponential and Logarithmic Functions]] | - exponential and logarithmic functions | - derivative of logarithmic and exponential functions | ## Differentiation with exponential functions - Apply the rules of differentiation to find the derivative of $f(x)=ke^{ax}$, where k and a are constants - Use the chain rule to prove $\frac{d}{dx}(e^{ax+b})=ae^{ax+b}$, where a and b are constants and $a\neq0$ - 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 $\frac{d}{dx}(\log_ax)=\frac{1}{x\ln{a}}$, 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