Differentiation and applications.
Build reliable derivative rules, explain the shape of a curve and use calculus to solve optimisation, growth and motion problems.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Review of derivatives and polynomial rules
Reconnect a derivative with its gradient meaning and differentiate polynomial and power expressions accurately.
- 02
Product rule
Differentiate a product by accounting for the change in both factors.
- 03
Quotient rule
Differentiate ratios while preserving their domains and signs.
- 04
Chain rule
Differentiate a function inside another function and keep the inner rate.
- 05
Natural exponential function and its derivative
Use the special rate property of the natural exponential function.
- 06
Exponential growth and decay
Interpret continuously changing exponential models through their derivatives.
- 07
Derivatives of sine/cosine/tangent
Differentiate trigonometric functions using radians and the inner rate.
- 08
Second derivatives and concavity
Use a second derivative to describe how the gradient changes, rather than confusing concavity with increasing or decreasing height.
- 09
Inflection points
Find inflection points by proving a change of concavity, including cases where a simple zero test fails.
- 10
Stationary-point classification
Classify stationary points using sign changes or a valid second-derivative test.
- 11
Curve analysis
Combine intercepts, limits and derivative information into a coherent curve sketch.
- 12
Optimisation
Turn a practical objective and its constraints into a one-variable calculus problem.
- 13
Rates and motion applications
Use derivatives of position and other quantities to explain rates, directions and changing speeds.
Topic review
Revisit the key ideas, then choose a practice paper.