Concepts · 3 min read
Calculus Basics: Derivatives and Integrals With Numbers
Derivative: slope at a point
Take two nearby points and shrink the gap h. The slope (f(3 + h) − f(3)) / h for f(x) = x² gets closer to 6:
| h | (f(3+h) − f(3)) / h |
|---|---|
| 1 | 7 |
| 0.1 | 6.1 |
| 0.01 | 6.01 |
| 0.001 | 6.001 |
| 0.0001 | 6.0001 |
The power rule gives it exactly: d/dx(xⁿ) = n·xⁿ⁻¹, so d/dx(x²) = 2x = 6 at x = 3.
Integral: area under a curve
Slice the area under y = x² from 0 to 3 into N rectangles and add them up. More slices means a better estimate:
| N slices | Sum of areas |
|---|---|
| 10 | 7.695 |
| 100 | 8.86545 |
| 1,000 | 8.9865 |
| 10,000 | 8.99865 |
The exact integral is x³/3 evaluated from 0 to 3: 27/3 = 9.
How they connect
The fundamental theorem of calculus says differentiation and integration are inverse operations: integrating a slope function recovers the original function (up to a constant).
Try it
Plot x² and a slope-line in the graphing calculator and check values with the scientific calculator. Related reading: what is e?
Frequently asked questions
Do I need limits to understand derivatives?
The numerical tables show the idea of a limit: as h shrinks the slope settles on a value.
What is calculus used for?
Rates of change and accumulation: speed and distance, growth, optimisation, areas and volumes.
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