Derivatives

The Derivative as a Limit

Imagine glancing at a car's speedometer: it reads 60 mph. That's the instantaneous speed — the rate of change of position at that exact moment. But how is it actually computed? The speedometer estimates it by measuring how far the car travels in a tiny slice of time and dividing. The smaller the slice, the more "instantaneous" the result.

This is precisely the idea behind the derivative. The average rate of change of $f$ over the interval $[x,\, x+h]$ is:

Geometrically this is the slope of the secant line through the two points $(x,\, f(x))$ and $(x+h,\, f(x+h))$. As we shrink $h$ toward zero, those two points slide together, and the secant line rotates toward a unique limiting position: the tangent line. Its slope is the derivative:

This limit is the formal definition of the derivative. If it exists at $x = a$, we say $f$ is differentiable at $a$. Notice that $f'$ is itself a function — it assigns to each $x$ the slope of $f$ at that point.

Let's compute one from scratch. Take $f(x) = x^2$:

So the derivative of $x^2$ is $2x$. At $x = 3$, the parabola has slope $6$. At $x = 0$, it has slope $0$ — the tangent is horizontal at the vertex.

You'll encounter several notations, all meaning the same thing: $f'(x)$, $\;\dfrac{df}{dx}$, $\;\dfrac{d}{dx}[f]$, and $\dot{f}$ (the dot notation is common in physics).

Derivative Graphs

Once you understand that the derivative is slope, you can extract a remarkable amount of information about $f$ just by looking at its derivative graph — no algebra required.

The sign of $f'(x)$ tells you whether $f$ is rising or falling. Where $f'(x) > 0$, the tangent tilts upward — $f$ is increasing. Where $f'(x) < 0$, it tilts downward — $f$ is decreasing. Where $f'(x) = 0$, the tangent is horizontal — $f$ momentarily flattens out. The magnitude $|f'(x)|$ tells you how steep: a large $|f'|$ means the function is racing up or down; a value near zero means it's barely moving.

The second derivative $f''(x)$ is the derivative of $f'(x)$ — the rate at which the slope itself is changing. This encodes concavity:

• $f''(x) > 0$ — the slope is increasing, so $f$ curves upward (concave up, like a bowl). The graph lies above its tangent lines.
• $f''(x) < 0$ — the slope is decreasing, so $f$ curves downward (concave down, like a hill). The graph lies below its tangent lines.
• $f''(x) = 0$ and changes sign — an inflection point, where the bend direction reverses.

Optimization & Critical Points

One of the most powerful applications of derivatives is finding where a function achieves its maximum or minimum value — a process called optimization. Engineers minimize drag. Economists maximize profit. Physicists find paths of least time. The mathematical principle behind all of these is the same: at a peak or a valley, the tangent line is horizontal, so the slope is zero.

A critical point is any $x = c$ where $f'(c) = 0$ or $f'(c)$ does not exist. Every local maximum and minimum must be a critical point — but not every critical point is a maximum or minimum. You need a test to decide.

The First Derivative Test: examine the sign of $f'$ on both sides of $c$.

• $f'$ goes $+ \to -$ at $c$: local maximum — the function rises, then falls
• $f'$ goes $- \to +$ at $c$: local minimum — the function falls, then rises
• $f'$ doesn't change sign: neither — it's an inflection point with horizontal tangent (e.g., $f(x) = x^3$ at $x = 0$)

The Second Derivative Test gives a quicker answer when $f'(c) = 0$:

The intuition is clean: if the function is concave up at a flat point, you're at the bottom of a bowl — a minimum. If concave down, you're at the top of a hill — a maximum.