Measuring instant change
A car’s speedometer shows how fast you’re going right now, not the average for the trip. The derivative captures this idea: the instantaneous rate of change of a function.
Geometrically, the derivative at a point is the slope of the tangent line to the graph there.
Notation
| Notation | Read as |
|---|---|
| f′(x) | "f prime of x" (prime notation) |
| dy/dx | "dee y by dee x" (Leibniz notation) |
| d/dx [f(x)] | "the derivative of f(x)" |
Finding a derivative is called differentiation.
From average to instantaneous
The average rate of change between x and x + h:
[f(x + h) − f(x)]/h
Let h shrink to 0:
f′(x) = lim (h→0) [f(x + h) − f(x)]/h
A quick example
f(x) = x²
- [(x + h)² − x²]/h = (2xh + h²)/h = 2x + h
- As h → 0: f′(x) = 2x
So the slope of y = x² is 2x: at x = 3 the slope is 6; at x = −1 it’s −2.
Basic rules (preview)
| Function | Derivative |
|---|---|
| c (a constant) | 0 |
| x | 1 |
| xⁿ | nxⁿ⁻¹ |
| cf(x) | cf′(x) |
| f(x) + g(x) | f′(x) + g′(x) |
Example: f(x) = 3x⁴ − 5x² + 7x − 2 → f′(x) = 12x³ − 10x + 7
What the derivative tells you
| f′(x) | Graph of f |
|---|---|
| positive | increasing |
| negative | decreasing |
| zero | flat (possible max or min) |
| large | steep |
Where derivatives fail
A function isn’t differentiable at:
- a corner or cusp (like |x| at 0)
- a discontinuity
- a vertical tangent
Applications
- Velocity = derivative of position
- Acceleration = derivative of velocity
- Marginal cost = derivative of cost
- Growth rates in biology and economics
Worked examples
Example 1: Derivative of f(x) = x³? 3x².
Example 2: Slope of y = x² at x = 5? f′(5) = 10.
Example 3: f(x) = 4x − 9. f′(x)? 4 (a line has constant slope).
Common mistakes
- Confusing f(x) with f′(x).
- Forgetting that the derivative of a constant is 0.
- Treating dy/dx as a fraction too casually in early work.
Try it yourself
- Derivative of 5x²?
- Derivative of x⁴ − 3x + 2?
- Slope of y = x³ at x = 2?
- Is f(x) = |x| differentiable at 0?
Answers: 1) 10x 2) 4x³ − 3 3) 12 4) No (sharp corner).
Quick recap
- The derivative is the instantaneous rate of change and the slope of the tangent line.
- Differentiation uses limits; notation f′(x) or dy/dx.
- Power rule preview: d/dx xⁿ = nxⁿ⁻¹.