TheIBAxolotl
From average to instantaneous
If you drive 120 km in 2 hours, your average speed is 60 km/h. But the speedometer shows your speed at one instant, which changes from moment to moment. Calculus answers: how do we measure a rate of change at a single instant?
On a graph, the average rate of change between two points is the gradient of the chord joining them. To get the rate at one point, slide the second point closer and closer. The chord turns into the tangent, and its gradient is the derivative.
Limits, informally
A limit describes the value an expression approaches as its input gets close to some number, even if it cannot be evaluated exactly there. We write
\[\lim_{h \to 0} g(h) = L\]
to mean that \(g(h)\) gets as close to \(L\) as we like when \(h\) is small enough (but not zero). You can estimate limits from a table of values or by zooming in on a graph.
Worked example
Estimate the gradient of \(y = x^2\) at \(x = 2\).
The chord from \((2, 4)\) to \((2 + h, (2 + h)^2)\) has gradient
\[\frac{(2 + h)^2 - 4}{h} = \frac{4h + h^2}{h} = 4 + h.\]
| \(h\) | 1 | 0.1 | 0.01 | 0.001 |
|---|---|---|---|---|
| chord gradient | 5 | 4.1 | 4.01 | 4.001 |
As \(h \to 0\) the gradients approach 4, so the gradient of the curve at \(x = 2\) is 4. Repeating this at a general point gives \(2x\), the gradient function of \(x^2\).
The derivative
The derivative of \(f\) is a new function that outputs the gradient of \(f\) at every point. Two notations are common:
- \(f'(x)\) ("f dash of x"), used when the function has a name;
- \(\dfrac{dy}{dx}\) ("dee y by dee x"), which reminds you it is a small change in \(y\) over a small change in \(x\).
\(f'(3) = 7\) means: at \(x = 3\), the graph is rising 7 units of \(y\) per unit of \(x\).
Derivative as a rate of change
In context, the derivative has units "output units per input unit". If \(V(t)\) is the volume of water in a tank in litres after \(t\) minutes, then \(V'(t)\) is in litres per minute. A negative value means the tank is emptying. Interpreting the sign and units in words is a frequent exam request.
Common mistakes
- Confusing the derivative (a gradient) with the function value at a point.
- Thinking a limit requires substituting \(h = 0\) directly, which gives \(\frac{0}{0}\).
- Forgetting units or the sign when interpreting a rate.
- Reading a table of chord gradients and reporting the last entry instead of the value they approach.
Exam tips
- In tables, describe the trend ("approaches 4") rather than quoting a single row.
- On Paper 2, the GDC can give a numerical derivative at a point: use it to check.
- When interpreting \(f'(a)\), write a full sentence including the units and what is changing.