1. What does dy/dx give on a graph?
- Area under the graph
- Tangent gradient
- Vertical intercept
Check the answer and reasoning
Tangent gradient
The derivative is the limiting rate of change with respect to x.
MATH / AS level
Differentiation and applications
A derivative measures instantaneous rate of change and tangent gradient. A stationary point has zero first derivative, but classification needs a sign change or further reasoning.
Learning objectives
- Explain differentiation and applications using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
A derivative measures instantaneous rate of change and tangent gradient. A stationary point has zero first derivative, but classification needs a sign change or further reasoning.
Optimisation starts by expressing the target in one variable and stating its physical domain. Check boundary values when the allowed interval is closed.
Equations and relationships
Power rule for suitable real domains.
Worked example
Differentiate 3x² − 4x.
Follow the solution
Apply the power rule term by term: 3×2x − 4.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.
Check your understanding
2. Differentiate 3x² − 4x.
- 6x² − 4
- 3x − 4
- 6x − 4
Check the answer and reasoning
6x − 4
Apply the power rule term by term: 3×2x − 4.
3. Does f′(a)=0 always imply a maximum?
- No
- Yes
- Only for positive a
Check the answer and reasoning
No
A stationary point can be a minimum or a stationary point of inflection.