Advanced Calculus – Differentiation

This lesson comprises three (3) master classes focusing on:

• First derivative
• Second derivative
• Product, quotient and chain rule
• Differentiation of trigonometric functions
• Differentiation of exponential and logarithmic functions
• Stationary points
• Concavity

Content:

MA-C2.1

• Establish the formulae \( \frac{d}{dx}(\sin x)=\cos x \) and \( \frac{d}{dx}(\cos x)=−\sin x \) by numerical estimations of the limits and informal proofs based on geometric constructions
• Calculate derivatives of trigonometric functions
• Establish and use the formula \( \frac{d}{dx}(a^x)=(\ln a)a^x \)
  • using graphing software or otherwise, sketch and explore the gradient function for a given exponential function, recognise it as another exponential function and hence determine the relationship between exponential functions and their derivatives
• Calculate the derivative of the natural logarithm function \( \frac{d}{dx}(\ln x)=\frac{1}{x} \)
• Establish and use the formula \( \frac{d}{dx}(\log_ax)=\frac{1}{x\ln a} \)

MA-C2.2

• Apply the product, quotient and chain rules to differentiate 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 this syllabus, for example \( xe^x \), \( \tan x \), \( \frac{1}{x^n} \), \( x\sin x \), \( e^{−x} \sin x \) and \( f(ax+b) \)
  • use the composite function rule (chain rule) to establish that \( \frac{d}{dx} \left\{ e^{f(x)} \right\}=f^{\prime}(x)e^{f(x)} \)
  • use the composite function rule (chain rule) to establish that \( \frac{d}{dx} \left\{ \ln f(x) \right\}=\frac{f^{\prime}(x)}{f(x)} \)
  • use the logarithmic laws to simplify an expression before differentiating
  • use the composite function rule (chain rule) to establish and use the derivatives of \( \sin (f(x)) \), \( \cos (f(x)) \) and \( \tan (f(x)) \)

MA-C3.1

• Use the first derivative to investigate the shape of the graph of a function
  • deduce from the sign of the first derivative whether a function is increasing, decreasing or stationary at a given point or in a given interval
  • use the first derivative to find intervals over which a function is increasing or decreasing, and where its stationary points are located
  • use the first derivative to investigate a stationary point of a function over a given domain, classifying it as a local maximum, local minimum or neither
  • determine the greatest or least value of a function over a given domain (if the domain is not given, the natural domain of the function is assumed) and distinguish between local and global minima and maxima
• Define and interpret the concept of the second derivative as the rate of change of the first derivative function in a variety of contexts, for example recognise acceleration as the second derivative of displacement with respect to time
  • understand the concepts of concavity and point of inflection and their relationship with the second derivative
  • use the second derivative to determine concavity and the nature of stationary points
  • understand that when the second derivative is equal to 0 this does not necessarily represent a point of inflection

MA-C3.2

• Use any of the functions covered in the scope of this syllabus and their derivatives to solve practical and abstract problems
• Use calculus to determine and verify the nature of stationary points, find local and global maxima and minima and points of inflection (horizontal or otherwise), examine behaviour of a function as \( x \to \infty \) and \( x \to −\infty \) and hence sketch the graph of the function
• Solve optimisation problems for any of the functions covered in the scope of this syllabus, in a wide variety of contexts including displacement, velocity, acceleration, area, volume, business, finance and growth and decay
  • define variables and construct functions to represent the relationships between variables related to contexts involving optimisation, sketching diagrams or completing diagrams if necessary
  • use calculus to establish the location of local and global maxima and minima, including checking endpoints of an interval if required
  • evaluate solutions and their reasonableness given the constraints of the domain and formulate appropriate conclusions to optimisation problems