- 1. Rates of Change: the Gradient of a Curve
Distinguish the average rate of change (the gradient of a chord) from the instantaneous rate of change (the gradient of the tangent), and read the sign of a gradient from the shape of a curve.
- 2. Differentiation from First Principles
Form the chord gradient in increment notation and take the limit as the increment tends to zero to obtain the derivative of a polynomial or simple algebraic fraction.
- 3. The Power Rule and Standard Derivatives
Use the power rule to differentiate $ax^n$, differentiate polynomials term by term, and know that the derivative of a constant is zero.
- 4. Gradient at a Point and the Second Derivative
Substitute into the gradient function to find the gradient at a point, solve dy/dx = k to locate a given gradient, and find and interpret the second derivative.
- 5. The Product and Quotient Rules
Differentiate products and quotients of algebraic functions using the product and quotient rules, and check a product-rule answer by expanding.
- 6. Function of a Function: the Chain Rule
Recognise a composite function and differentiate it with the chain rule, including the shortcut for (ax + b) raised to a power and cases with a quadratic inside.
- 7. Maxima and Minima
Locate stationary points by solving dy/dx = 0, find the coordinates of a turning point, and determine its nature using the second derivative or a gradient-sign table.
- 8. Rate of Change, Velocity and Acceleration
Interpret the derivative as a rate of change, differentiate displacement to obtain velocity and acceleration, and form and optimise functions from worded problems.
