Introduction to Derivatives
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what a derivative represents, the difference between average and instantaneous rate of change, and why derivatives are one of the most useful ideas in all of mathematics.
The Concept
Section titled “The Concept”You already know how to find the slope between two points: rise over run. If a car travels 120 miles in 2 hours, its average speed is 60 mph. That’s the slope of the line connecting the start and end points on a distance-vs-time graph.
But what if you want to know how fast the car is going at one specific instant? Not over 2 hours, but right now, at this exact second? That’s a different question, and algebra can’t answer it. Calculus can.
The derivative of a function at a point is the instantaneous rate of change at that point. It’s the slope of the curve at a single location, not between two locations.
Average rate of change over an interval [a, b] is the slope of the line connecting two points on the curve:
Instantaneous rate of change at a single point x = a is the slope of the tangent line at that point. This is the derivative.
The blue curve is the function. The green line just touches the curve at the orange dot without crossing it. The slope of that green tangent line is the derivative at that point. It tells you exactly how steep the curve is right there.
We write the derivative of f at x using two common notations:
- f’(x) (read “f prime of x”) is Lagrange notation
- dy/dx (read “dee y dee x”) is Leibniz notation
Both mean the same thing: the instantaneous rate of change of y with respect to x.
Worked Example
Section titled “Worked Example”Example 1: Average vs. instantaneous
A ball is dropped, and its position after t seconds is s(t) = 5t² meters.
Average speed from t = 1 to t = 3:
That’s the average over 2 seconds. But the ball is accelerating, so it’s going slower at t = 1 and faster at t = 3. The derivative will tell us the exact speed at any single moment. (We’ll compute it properly in the next lesson using the limit definition.)
Example 2: Linear functions
For f(x) = 3x + 2, the derivative is 3 everywhere. A straight line has the same slope at every point, so the rate of change never varies. The tangent line at any point is the line itself.
Example 3: Why nonlinear functions are different
For f(x) = x², the rate of change is not constant. At x = 1, the curve is rising gently. At x = 5, it’s rising steeply. The derivative captures this: it gives a different number at each x-value, telling you exactly how fast the function is changing at that specific spot.
Real-World Application
Section titled “Real-World Application”Derivatives show up everywhere:
- Physics: velocity is the derivative of position, acceleration is the derivative of velocity. Newton’s second law (F = ma) is a statement about derivatives.
- Economics: marginal cost is the derivative of total cost. It tells you how much one more unit costs to produce.
- Biology: the rate of population growth at a specific time is a derivative.
- Medicine: how fast a drug’s concentration in the blood is changing is a derivative.
- Engineering: stress and strain rates in materials are derivatives.
Example: when a doctor reads your heart rate monitor, they’re looking at a derivative. The monitor shows how fast your heart is beating right now, not your average heart rate over the last hour.