Differentiation 2

Building on basic differentiation, this topic explores advanced techniques such as the chain rule, product rule, quotient rule, and implicit differentiation. Students will also apply differentiation to more complex curves and real-world optimization problems.

Learning Outcomes:
The learner should be able to:

• Determine derivatives of trigonometric functions from first principles.
• Use the product, quotient, and chain rules for differentiation.
• Apply various differentiation techniques to determine derivatives of trigonometric functions.
• Differentiate inverse trigonometric functions.
• Sketch exponential and logarithmic functions graphs to observe their shape and characteristics.
• Generate the derivative of the exponential function .
• Deduce the general formula for differentiating exponential functions.
• Relate exponential functions to natural logarithms, deduce logarithmic derivatives from exponential derivatives.
• Apply the natural logarithms to differentiate exponential functions.
• Generate Maclaurin’s polynomial expansion and use it to expand functions.
• Apply Maclaurin’s theorem in expansions to: i) approximate functions, ii) truncate series to desired degree.
• Identify key features (intercepts, asymptotes, range, domain) using knowledge of coordinate geometry and algebra.
• Analyse function behaviour (increasing/ decreasing, concavity, and turning points).
• Apply concepts of differentiation, coordinate geometry to accurately sketch curves.
• Sketch irrational curves.
• Develop graphing skills to enhance spatial reasoning and visualisation in solving real-world contexts such as optimisation.