Derivatives - Measuring Rates of Change
A derivative measures how fast a function is changing at any given point. Think of it as the instantaneous rate of change – not the average change over a long interval, but the exact rate at one specific instant. If a function describes a position, its derivative describes velocity. If the function describes velocity, its derivative describes acceleration.
The Gradient of a Curve
On a straight line, the gradient (slope) is constant and easy to compute. On a curve, the gradient is different at every point. The derivative at point x gives the gradient of the tangent line to the curve at that exact point. The tangent line just touches the curve at one point and has the same slope as the curve there.
The Formal Definition
The derivative of f at x is defined as the limit:
f′(x) = limh → 0 [f(x + h) − f(x)] / h
This is called the first principles definition. It calculates the slope of a secant line as the two points move infinitely close together.
Notation
The derivative of y = f(x) can be written as:
f′(x) or dy/dx or y′ or d/dx [f(x)]
All four mean exactly the same thing.
Differentiation Rules
| Rule | Formula | Example |
|---|---|---|
| Power Rule | d/dx [xn] = nxn−1 | d/dx [x5] = 5x4 |
| Constant Rule | d/dx [c] = 0 | d/dx [7] = 0 |
| Constant Multiple | d/dx [cf(x)] = cf′(x) | d/dx [3x²] = 6x |
| Sum / Difference | d/dx [f ± g] = f′ ± g′ | d/dx [x³ + x] = 3x² + 1 |
| Product Rule | d/dx [fg] = f′g + fg′ | d/dx [x² sin x] = 2x sin x + x² cos x |
| Quotient Rule | d/dx [f/g] = (f′g − fg′) / g² | d/dx [sin x / x] = (x cos x − sin x) / x² |
| Chain Rule | d/dx [f(g(x))] = f′(g(x)) · g′(x) | d/dx [(x²+1)3] = 3(x²+1)² · 2x |
Derivatives of Standard Functions
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| ex | ex |
| ln x | 1/x |
| xn | nxn−1 |
Worked Examples
Apply the power rule term by term:
f′(x) = 12x² − 14x + 2.
dy/dx = 3x² − 3. At x = 2: 3(4) − 3 = 12 − 3 = 9.
Outer function: u4, derivative = 4u3. Inner function: 2x + 1, derivative = 2.
dy/dx = 4(2x + 1)3 × 2 = 8(2x + 1)3.
dy/dx = 3x² − 12x + 9. Set equal to zero:
3(x² − 4x + 3) = 0 → 3(x − 1)(x − 3) = 0.
Stationary points at x = 1 and x = 3.
Second Derivative
The second derivative f″(x) or d²y/dx² is the derivative of the derivative. It measures how the rate of change is itself changing – in other words, the concavity of the curve.
If f″(x) > 0 at a stationary point: local minimum.
If f″(x) < 0 at a stationary point: local maximum.
Key Takeaways
- The derivative gives the instantaneous rate of change and the slope of the tangent line.
- Power rule: d/dx [xn] = nxn−1.
- Product, quotient, and chain rules handle more complex expressions.
- Stationary points occur where f′(x) = 0; the second derivative classifies them.
Practice Questions
- Differentiate f(x) = 6x4 − 3x² + 8.
- Find the gradient of y = 2x³ + 5x at x = −1.
- Differentiate y = (x² + 3)(2x − 1) using the product rule.
- Use the chain rule to differentiate y = sin(3x²).
- Find the stationary points of y = x³ − 12x and classify each using the second derivative.