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What is a Derivative?

A derivative measures the instantaneous rate of change of a function — how fast it's changing at a specific moment. It's the slope of a curve at a single point. The derivative is arguably the most powerful single idea in all of mathematics.

In this lesson

  1. Rate of change — the core idea
  2. Derivative as slope
  3. The formal definition
  4. Notation: f'(x), dy/dx, and more
  5. Real-world meaning

1 Rate of Change — The Core Idea

You already understand rate of change intuitively. Speed is rate of change of position — 60 mph means your position changes by 60 miles per hour. Acceleration is rate of change of speed. The steepness of a hill is the rate of change of elevation with respect to horizontal distance.

Average rate of change between two points is easy: (change in output) / (change in input) = Δy/Δx. For a car traveling 120 miles in 2 hours, average speed = 120/2 = 60 mph.

But what if speed varies? What is your speed at exactly 1:32 PM — not over an hour, not over a minute, but at a single instant? That instantaneous rate of change is what the derivative computes.

The Key Insight

A derivative converts 'average rate of change over an interval' into 'instantaneous rate of change at a point' by shrinking the interval to zero using a limit.

2 Derivative as Slope

On a graph, the average rate of change between two points is the slope of the line connecting them (a secant line). As you move the second point closer and closer to the first, the secant line rotates and approaches the tangent line — a line that just touches the curve at one point.

The derivative at a point is the slope of the tangent line at that point. For a straight line y = mx + b, the slope is constant everywhere — the derivative is m at every point. For a curve like y = x², the slope changes at every point, and the derivative tells you exactly what that slope is.

3 The Formal Definition

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

This is the difference quotient taken to its limit. [f(x+h) − f(x)] / h is the slope of the secant line between x and x+h. As h→0, this approaches the slope of the tangent at x.

Derivative from the Definition
Find f'(x) for f(x) = x²
1Write the difference quotient: [f(x+h) − f(x)] / h = [(x+h)² − x²] / h
2Expand: [x² + 2xh + h² − x²] / h = [2xh + h²] / h
3Simplify: h(2x + h) / h = 2x + h
4Take the limit as h→0: lim(h→0)(2x + h) = 2x
Answer: f'(x) = 2x — the derivative of x² is 2x

This means: at any point x on the parabola y=x², the slope of the tangent line is 2x. At x=3, the slope is 6. At x=−1, the slope is −2. At x=0 (the vertex), the slope is 0.

4 Derivative Notation

Several notations exist for the derivative, all meaning the same thing:

Common derivative notations
f'f'(x) — Lagrange notation. "f prime of x." Most common in pure math courses.
dy/dxdy/dx — Leibniz notation. "dee y dee x." Emphasizes the ratio of changes. Common in physics and engineering.
ẋẋ (dot notation) — Newton's notation. Used primarily in physics for time derivatives.
DfDf — operator notation. Common in advanced mathematics.

5 Real-World Meaning

Physics: velocity is the derivative of position with respect to time. Acceleration is the derivative of velocity. Force equals mass times acceleration — Newton's second law is fundamentally about derivatives.

Economics: marginal cost is the derivative of total cost with respect to quantity — the cost of producing one more unit. Marginal revenue is the derivative of revenue. Profit is maximized where marginal cost = marginal revenue, which is a derivative condition.

Medicine: pharmacokinetics models how drug concentration in blood changes over time using derivatives. The rate at which a drug clears the body is described by a differential equation — an equation involving derivatives.

Machine learning: neural networks are trained by minimizing a loss function using gradient descent — moving in the direction of the negative derivative (gradient) to find the minimum. Every AI model you've ever used was trained with derivatives.

Practice Problems

If f(x) = x³, use the definition to find f'(x) at x=2. (Hint: the derivative of x³ is 3x²)
f'(x) = 3x². At x=2: f'(2) = 3(4) = 12. The tangent to y=x³ at x=2 has slope 12.
A car's position is s(t) = 4t² meters at time t seconds. What is its velocity at t=3?
Velocity = s'(t). The derivative of 4t² is 8t. At t=3: v = 8(3) = 24 m/s.
Where is the slope of y = x² equal to zero?
f'(x) = 2x = 0 → x = 0. The tangent line is horizontal at x=0, the vertex of the parabola.