Single-variable calculus from first principles
Computing a derivative from the limit definition every time would be exhausting. A handful of rules let you differentiate almost anything by inspection. Learn these and you'll never need the limit again for ordinary functions.
The workhorse. To differentiate a power, bring the exponent down in front and drop it by one:
Measuring a car's speed by hand, marking distances and timing them with a stopwatch, works but is painfully slow. A speedometer wired to the wheels gives you the same answer instantly. The derivative rules are that wired-in shortcut: instead of grinding through the limit definition every time, you read the slope straight off the formula.