The Derivative

01From secant to tangent

Pick two points on a curve, $(a, f(a))$ and $(a+h,\, f(a+h))$. The line through them — a secant line — has slope

$$\frac{f(a+h) - f(a)}{h}$$

which is the average rate of change of $f$ over the interval. Now slide the second point toward the first by letting $h \to 0$. The secant lines pivot into a single limiting line — the tangent line — and their slopes converge to the instantaneous rate of change. That sliding picture is the hero graph above, and it is the entire conceptual content of the derivative. Everything else is bookkeeping.

02The definition

The derivative of $f$ at $a$ is

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

provided the limit exists. There's an equivalent form that's sometimes easier to use and to recognize on exams:

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Notice this is always a $\frac{0}{0}$ indeterminate form — the numerator and denominator both vanish — which is why every Module 02 technique (expand, factor, cancel, conjugate) reappears here. The derivative is not new machinery; it is a specific limit you already know how to compute.

03Two worked computations

Example 1: $f(x) = x^2$.

$$f'(x) = \lim_{h\to 0}\frac{(x+h)^2 - x^2}{h} = \lim_{h\to 0}\frac{2xh + h^2}{h} = \lim_{h\to 0}\,(2x + h) = 2x$$

Example 2: $f(x) = \dfrac{1}{x}$. Combine the fractions in the numerator first:

$$f'(x) = \lim_{h\to 0}\frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h\to 0}\frac{\frac{-h}{x(x+h)}}{h} = \lim_{h\to 0}\frac{-1}{x(x+h)} = -\frac{1}{x^2}$$

The rhythm is always: set up the difference quotient, do algebra until the lone $h$ in the denominator cancels, then let $h \to 0$.

04The derivative as a function, and notation

Computing $f'$ at a general $x$ (as above) produces a new function $f'(x)$ whose value at each point is the slope there. You'll see several notations for the same object, and you need to read all of them fluently:

$$f'(x) \qquad y' \qquad \frac{dy}{dx} \qquad \frac{d}{dx}f(x)$$

Lagrange's prime notation is compact; Leibniz's $\frac{dy}{dx}$ makes the "change in $y$ per change in $x$" meaning visible and will shine in the chain rule and related rates. Evaluation at a point is written $f'(3)$ or $\left.\frac{dy}{dx}\right|_{x=3}$.

05What the derivative means

Interpretation is half the course. If $s(t)$ is position at time $t$, then $s'(t)$ is velocity — the speedometer reading at instant $t$, not an average over a trip. If $C(x)$ is the cost of producing $x$ units, $C'(x)$ is marginal cost, roughly the cost of one more unit. In every case the units of $f'$ are (units of output) per (unit of input): a population in thousands measured over years has a derivative in thousands per year. Exam tip: when asked to "interpret $f'(5) = 12$," answer with the rate, the units, and the instant — "at time 5, the quantity is increasing at 12 units per hour."

06The tangent line equation

The tangent line to $y = f(x)$ at $x = a$ passes through $(a, f(a))$ with slope $f'(a)$, so point–slope form gives

$$y = f(a) + f'(a)\,(x - a)$$

Example: for $f(x) = x^2$ at $a = 3$: $f(3) = 9$, $f'(3) = 6$, so $y = 9 + 6(x - 3) = 6x - 9$. This little formula is also the linear approximation of $f$ near $a$ — the best straight-line stand-in for the curve — an idea that returns throughout calculus.

07Where derivatives fail to exist

The defining limit can fail, and the failures have geometric signatures:

• Corner — one-sided slopes disagree. $f(x) = |x|$ at $0$: slope $-1$ from the left, $+1$ from the right.
• Cusp — slopes blow up to $+\infty$ on one side and $-\infty$ on the other, like $x^{2/3}$ at $0$.
• Vertical tangent — the tangent exists but is vertical (infinite slope), like $x^{1/3}$ at $0$.
• Discontinuity — no continuity, no derivative. Ever.

That last item is a theorem worth stating: differentiable $\Rightarrow$ continuous. The converse is false — $|x|$ is the standard counterexample — so "continuous but not differentiable" is a phrase to keep handy.

08Reading derivatives from graphs

Given only a graph of $f$, you can sketch $f'$: where $f$ rises, $f' > 0$; where $f$ falls, $f' < 0$; at smooth peaks and valleys, $f' = 0$. Steepness translates to magnitude. This graphical fluency — matching a function's graph to its derivative's graph — is a favorite exam question and builds the intuition that Module 08 (curve sketching) will formalize. When you only have a table of values, estimate $f'(a)$ with the symmetric difference quotient $\frac{f(a+h) - f(a-h)}{2h}$ using the nearest available points.