Analysis & Calculusconceptintermediate

The Derivative — Slope at an Instant

“A derivative is just 'the slope at this very instant'.”

The formula

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

How to read it: the limit of how many times more the output changes than a tiny change in x

h: — a tiny change in x
f(x+h)−f(x): — how much the function value changed
lim(h→0): — the limit as that change shrinks toward 0

The hook

The derivative looks scary, but it's really just one thing — 'how steep is it right at this instant'.

In plain words

Take the slope between two points (rise ÷ run), then slide the two points infinitely close to get 'the slope at a single point'.

The intuition

Rest a ruler against one point of a curve and it becomes the tangent line. Its slope is the derivative — like the 'current steepness' of a hill.

How it's built

The top f(x+h)−f(x) is the 'rise', the bottom h is the 'run' → their ratio is a slope. Send h to 0 and what remains is the slope at that instant.

Example

For f(x)=x² at x=1: (1+h)²−1 = 2h+h², divide by h to get 2+h, and as h→0 that's 2. So the slope at x=1 is 2.

Common mistake

'Instantaneous change' does not mean the change is 0 — the ratio of two near-zero quantities can converge to a nonzero value.

Where it's used

Velocity (derivative of position), optimization (max/min where slope is 0), and gradient descent in machine learning — all of it is this one idea.

Where it came from

Newton and Leibniz each invented it in the 1600s; Leibniz's notation dy/dx is the one that survived.

Prerequisites

Limits

Quick check

For f(x)=x², what is the derivative (slope) at x=3?

3
6✓
9
0

Practice

For f(x)=x³, what is the slope at x=2?

Answer: 12

Solution:

f'(x)=3x²
Plug in x=2 → 3·4 = 12

Takeaway: Derivative = slope at that point. Set up the formula, substitute once.

On the curve y=x², where is the tangent's slope 0? Also jot down why.

Answer: undefined