Differentiating gives the rate of change of a function: the gradient of the curve at a point. That interpretation matters, because exam questions rarely stop at the derivative — they ask for tangents, stationary points, or the maximum of a modelled quantity.
Reading the structure first
| Structure of the expression | Rule |
|---|---|
| A sum of powers of x | Differentiate term by term |
| A function inside another function | Chain rule |
| Two functions multiplied | Product rule |
| One function divided by another | Quotient rule (or rewrite as a product) |
| x and y mixed together | Implicit differentiation |
| x and y both given in terms of t | Parametric: divide dy/dt by dx/dt |
Where marks are actually lost
- Forgetting the inner derivative in the chain rule.
- Getting the order of terms wrong in the quotient rule numerator.
- Differentiating trigonometric functions with the angle in degrees rather than radians.
- Finding stationary points but not classifying them when the question asks for a maximum or minimum.
Applications you should expect
- Equation of a tangent or normal at a given point.
- Stationary points, classified using the second derivative or a sign change.
- Increasing and decreasing intervals.
- Optimisation in context, where the final answer must be interpreted in the units of the question.
On optimisation questions, always answer the question that was asked. A derivative set to zero gives a value of x; the marks are often for the maximum volume, cost or area that follows from it.