The Derivative

A slope for every point

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ is the slope of $f$ at $x$.
• Lines: $f'$ is the constant slope. Constants: $f' = 0$. $(x^2)' = 2x$ and $(x^3)' = 3x^2$.
• $f' > 0$ where $f$ is increasing, $f' < 0$ where it's decreasing, and $f' = 0$ at peaks and valleys.

Steps

1. Spot the pattern
2. The derivative function
3. Use the definition
4. Constant
5. Slope of a cubic
6. Cube it