Differentiation: local rate of change
Connect the gradient of a curve to the rate at a particular point.
Before you begin
- Functions and gradients
- Index laws
Distinguish average and instantaneous rates
An average rate uses the change between two distinct inputs. An instantaneous rate describes the limiting gradient as those inputs approach each other. The derivative f′(x) gives that local gradient where it exists. Its units are output units per input unit; for displacement against time, the derivative is velocity.
Differentiate powers and inspect stationary points
For a polynomial term axⁿ, the derivative is anxⁿ⁻¹; a constant has derivative zero. Differentiate each term separately. A stationary point has derivative zero, but it need not be a maximum or minimum. Inspect the derivative on either side or use further analysis to determine the behaviour.
Worked example
For f(x) = x³ − 3x, find the gradient at x = 2.
Show the worked solution
- Differentiate term by term: f′(x) = 3x² − 3.
- Substitute x = 2 into the derivative, not the original function.
- Calculate 3 × 4 − 3 = 9.
Gradient 9.
Try it yourself
What is the derivative of 4x³ + 7?
Common mistakes
- Use the original function to find the point's y-coordinate.
What can you explain now?
For f(x) = x² − 6x + 5, locate the stationary point and classify it.
Compare with the explanation
(3, −4), a minimum.
f′(x) = 2x − 6 is zero at x = 3. The original function gives −4. The derivative changes from negative to positive across 3, so the curve turns upward.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗