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What is a Derivative?

The Derivative

The rate of change at an instant

The derivative answers one of the most important questions in mathematics: What is the rate of change at a single point?

Think about driving a car. Your speedometer shows your instantaneous speed - not your average speed over the trip, but exactly how fast you're going RIGHT NOW. That's what a derivative tells us.

The Derivative Definition
The limit of the difference quotient - measuring instantaneous rate of change

Illustration: derivative-definition

💡
The Speedometer Analogy

Imagine driving from home to work:
- Average speed = total distance / total time (easy to calculate)
- Instantaneous speed = speed at exactly 8:15:23 AM (how do we find this?)

The derivative is the mathematical tool that lets us find instantaneous rates of change.

Derivative as a Limit
Derivative
f'(x) = lim[h->0] [f(x+h) - f(x)] / h

"f prime of x"

The derivative is the limit of the average rate of change as the interval shrinks to zero.
Finding a Derivative from the Definition
Medium
Find f'(x) if f(x) = x^2
1
Write the definition
f'(x) = lim[h->0] [f(x+h) - f(x)] / h
2
Substitute f(x) = x^2
= lim[h->0] [(x+h)^2 - x^2] / h
3
Expand (x+h)^2
= lim[h->0] [x^2 + 2xh + h^2 - x^2] / h
4
Simplify
= lim[h->0] [2xh + h^2] / h = lim[h->0] h(2x + h) / h
5
Cancel and evaluate
= lim[h->0] (2x + h) = 2x
Answer: f'(x) = 2x
💡 This proves the power rule for n=2: d/dx[x^2] = 2x

Practice

1. The derivative represents:
Average rate of change
Instantaneous rate of change
Total change
Area under curve
2. Using the definition, find f'(x) for f(x) = 3x
3x
3
0
x
Key Takeaways

  • The derivative gives the instantaneous rate of change

  • It's defined as a limit of average rates

  • Notation: f'(x) or dy/dx or d/dx[f(x)]

  • Geometrically: the slope of the tangent line