VG2
kalkulus
Quiz
What is differentiation?
Differentiation finds the instantaneous rate of change of a function — how fast something is changing at a specific point.
🎮 Try it yourself — interactive!
See the tangent line move along the curve and the derivative calculated live.
→ Open interactive parabola (with tangent)
See the tangent line move along the curve and the derivative calculated live.
→ Open interactive parabola (with tangent)
The intuitive idea
A speedometer shows your speed right now — not your average speed for the whole trip. This is the essence of differentiation: the instantaneous rate of change.
Notation
f'(x) or dy/dx
Differentiation rules
| f(x) | f'(x) | Example |
|---|---|---|
| c (constant) | 0 | f(x)=7 → 0 |
| x | 1 | f(x)=x → 1 |
| xⁿ | n·xⁿ⁻¹ | f(x)=x³ → 3x² |
| ax+b | a | f(x)=3x+2 → 3 |
Step-by-step example
Differentiate f(x) = 4x³ + 2x² - 5x + 1
f'(x) = 4·3x² + 2·2x - 5 + 0
f'(x) = 12x² + 4x - 5
f'(x) = 12x² + 4x - 5
What is differentiation used for?
- Finding maximum and minimum points (where f'(x) = 0)
- Analysing whether a function is increasing or decreasing
- Speed and acceleration in physics
- Optimisation in economics
Power rule memory trick:
Bring down the exponent and subtract 1:
xⁿ → n·xⁿ⁻¹
Bring down the exponent and subtract 1:
xⁿ → n·xⁿ⁻¹
Nature is written in mathematical language, and the letters are triangles, circles and other geometrical figures.
— Galileo Galilei (1564–1642)
🧠 Test yourself
Question 1 of 3
What is the derivative of f(x) = x3?