10.1110 min read

The Derivative

The derivative of $f$ at $x_0$ is the limit of the average rate of change as the interval shrinks to zero: $f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}.$ It is a single number — the slope of the tangent line to the graph of $f$ at the point $(x_0, f(x_0))$ .

Geometrically: the derivative is the slope of the "best straight-line approximation" to the graph near $x_0$ . If $f'(x_0) > 0$ , the function is increasing at $x_0$ . If $f'(x_0) < 0$ , it is decreasing. If $f'(x_0) = 0$ , the tangent is horizontal — a candidate for a local extremum.

The derivative measures sensitivity: how much does $f$ change for a tiny change in input? A large $|f'(x_0)|$ means small changes in $x$ cause large changes in $f$ (sensitive). A small $|f'(x_0)|$ means $f$ is relatively flat (insensitive).

Formal View

Definition 10.10 — The Derivative

The derivative of

f

x_0

f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h},

provided this limit exists. We say

f

is differentiable at $x_0$ when the derivative exists.

The derivative is the slope of the tangent line to the graph of $f$ at $x_0$ . It equals the best linear approximation rate.

Interactive Visualization

Secant → Tangent Line

Why This Matters

The derivative is the central object of calculus — it enables optimization, rates of change, and local approximation.

Gradient computation in neural networks: the derivative of the loss w.r.t. each weight guides gradient descent.
Physics: the second derivative of position w.r.t. time is acceleration ( $F = ma$ ).
Finance: derivatives (financial) are named after this: their price depends on the derivative of the underlying.

Learning Resources

Derivative intro

Khan Academy

Introduction to the derivative as a limit of average rates of change.

10 min

Essence of calculus, Chapter 2

3Blue1Brown

Geometric and intuitive explanation of the derivative.

17 min

Quiz

Question 1

The derivative $f'(x_0)$ is:

Question 2

If $f'(x_0) > 0$ , the function $f$ is increasing at $x_0$ .

Common Mistakes

Thinking the derivative equals $f(x_0)/x_0$ — it is a LIMIT of differences, not a ratio of the function to its input.
Confusing $f'(x_0)$ (a number, the slope at one point) with the derivative function $f'(x)$ (a function of $x$ ).