What Is a Derivative? The Mathematics of Instantaneous Change

A derivative measures how fast something is changing at a single instant.

Here's the unlock: you already know what a derivative means — you just call it something else. When your speedometer reads 60 mph, that's a derivative. It's not your average speed over the trip. It's how fast you're going right now, at this exact moment.

The derivative is instantaneous velocity generalized. It answers the question: if this quantity keeps changing at its current rate, how much would it change in the next tiny instant?

Calculus didn't invent the concept. It invented the mathematics to calculate it.

The Problem: Slope at a Point

Slope is "rise over run" — how much y changes when x changes.

Between two points (x₁, y₁) and (x₂, y₂):

slope = (y₂ - y₁) / (x₂ - x₁)

But what if you want the slope at a single point? You can't divide by zero. You need two points to have a rise and a run.

Yet curves clearly have steepness at each point. A ball rolling down a hill has a speed at each instant. The question "how fast?" has an answer, even at a single moment.

Calculus solves this paradox.

The Solution: Take a Limit

Start with two points that are close together. Calculate the slope between them.

Now move the points closer. Calculate again.

Keep doing this — closer and closer. Watch what the slope approaches.

The derivative is what the slope approaches as the two points become infinitely close.

You never actually reach "one point." But you can get arbitrarily close, and the slope converges to a specific value. That value is the derivative.

The Formal Definition

For a function f(x), the derivative at point x is:

f'(x) = lim[h→0] (f(x+h) - f(x)) / h

Here's what this means:

• Start at x, where the function value is f(x)
• Move a tiny distance h to x+h, where the function value is f(x+h)
• The slope between these points is (f(x+h) - f(x)) / h
• Take the limit as h approaches 0

The result is the instantaneous slope at x.

Notation

Several notations exist for derivatives:

f'(x) — "f prime of x" (Lagrange notation)

dy/dx — "dee y dee x" (Leibniz notation, treating the derivative as a ratio of infinitesimals)

df/dx — same as above, with function name

d/dx f(x) — the operator "d/dx" applied to f

All mean the same thing: the derivative of the function with respect to x.

Example: f(x) = x²

Let's calculate the derivative of f(x) = x² using the definition.

f'(x) = lim[h→0] (f(x+h) - f(x)) / h = lim[h→0] ((x+h)² - x²) / h = lim[h→0] (x² + 2xh + h² - x²) / h = lim[h→0] (2xh + h²) / h = lim[h→0] (2x + h) = 2x

So the derivative of x² is 2x.

At x = 3, the slope is 2(3) = 6. At x = -1, the slope is 2(-1) = -2.

The derivative gives you the slope at every point as a function of x.

Geometric Interpretation

The derivative at a point is the slope of the tangent line at that point.

A tangent line touches the curve at exactly one point (locally) and has the same slope as the curve there.

The process of finding the derivative is equivalent to:

1. Draw a secant line through two nearby points
2. Slide one point toward the other
3. The secant approaches the tangent
4. The secant's slope approaches the tangent's slope

The derivative captures the tangent line's slope.

Physical Interpretation

If f(t) is position at time t, then f'(t) is velocity.

If f(t) is velocity at time t, then f'(t) is acceleration.

The derivative is the rate of change. It tells you how fast the quantity is changing at each moment.

Speed is the derivative of position. Acceleration is the derivative of velocity (second derivative of position).

When Derivatives Don't Exist

Not every function has a derivative everywhere.

Corners: The absolute value function |x| has a corner at x = 0. The left slope is -1, the right slope is +1. No single tangent line exists.

Cusps: Sharp points where the curve reverses direction infinitely steeply.

Discontinuities: If the function jumps, there's no meaningful slope at the jump.

Vertical tangents: If the tangent line is vertical, the slope is undefined.

A function is differentiable at a point if its derivative exists there.

Derivative as a Function

The derivative of f(x) is itself a function f'(x).

f(x) = x² has derivative f'(x) = 2x

The original function tells you the value at each point. The derivative tells you the slope at each point.

You can differentiate again: f''(x) is the second derivative — the rate of change of the rate of change.

Why "Instantaneous" Is Tricky

Strictly speaking, nothing happens "at an instant." Change requires duration.

The derivative resolves this by asking: what value does the average rate of change converge to as the duration shrinks?

It's not that change happens in zero time. It's that the rate has a well-defined limit as time intervals approach zero.

This is the genius of limits: extracting meaningful values from impossible-seeming questions.

The Core Insight

The derivative is the slope at a point, computed through limits.

You can't directly calculate rise/run with only one point. But you can see what slope the curve is approaching as two points converge. That limit is the instantaneous rate of change.

Every speedometer reading, every growth rate, every marginal cost — these are derivatives. Calculus gave us the tools to calculate them precisely.

When you take a derivative, you're asking: how fast is this changing, right now? The answer is a number that captures the curve's steepness at a single location.

Part 1 of the Calculus Derivatives series.

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