Change — A Daily Idea
Change is not a new idea introduced by mathematics — it is something we already describe constantly, in ordinary language, often without any number attached to it. We say traffic is “building up,” a fever is “coming down,” someone’s mood is “swinging,” a market is “heating up.” Each of these is a statement about one quantity moving as another quantity — usually time — moves along with it.
The language is loose, the rate is left unspecified, and “how fast” is rarely the point of the sentence. Mathematics enters only when this loose daily sense of change is asked to become precise: not just that something changed, but by how much, and how quickly, at a given moment.
- Traffic is building up near the signal.
- Her fever is finally coming down.
- The market has been heating up all week.
- His mood swings within the same hour.
- The room is cooling down slowly.
- Sales pick up sharply during the festival season.
- The patient’s pulse is dropping.
- Prices are rising faster than last month.
- The battery is draining quickly today.
- Her interest in the subject grew steadily over the term.
Why “Rate at a Point” is an Important Idea?
An ordinary rate is easy to picture: a car covers 100 km in 2 hours, so its rate is 50 km/hr. But that number is an average — it says nothing about what the car was doing at any single instant within those two hours. It may have been stopped at one moment and speeding at another.
The harder question, and the one that took mathematics a very long time to answer properly, is: what is the rate at one exact instant — at a single point in time, which by itself has no duration to measure a rate over?
This is not a minor technicality. A rate, by its very definition, needs two things to compare — a “before” and an “after,” a stretch of time or space across which something changed. A single instant has neither. Asking for a rate at a point sounds, at first, like asking for the speed of something frozen in time.
The way out of this is exactly the machinery already built while probing functions: instead of asking for the rate at the point, ask what the rate is heading toward as the surrounding interval is shrunk closer and closer to that point — left side and right side both approaching, the same way a limit was built earlier.
If both sides settle on the same number as the interval shrinks toward zero width, that settled number is taken to be the rate at the point, even though no actual interval of zero width was ever used to compute it.
Historically, this is the same instinct behind ideas like instantaneous energy release or instantaneous force — physical quantities that seem to require a “this very moment” measurement, resolved the same way: not by measuring at a single instant directly, but by observing what a small interval around that instant approaches as it is shrunk down, $x$ to $x + \epsilon$, with $\epsilon$ taken smaller and smaller.
The Five Primitive Functions
Before any formula is written down, each of the five primitive forms can be asked the same conceptual question: as $x$ increases a little, does $y$ change at all, and if so, does it change by the same amount everywhere, or does the amount of change itself depend on where you are?
Constant — $f(x) = k$ The output never moves, no matter how $x$ moves. There is nothing here for a rate of change to describe — the rate of change of a constant function is zero, everywhere, by the nature of the function itself, before any limit is even computed.
Linear / Power — $f(x) = ax^b + c$ When $b = 1$, the function changes by the same fixed amount for every equal step in $x$ — this is exactly what makes a straight line “straight.” When $b \neq 1$, the amount of change itself depends on where $x$ currently is — near $x = 0$ a power function may barely move, while far from $0$ the same step in $x$ produces a much larger change in $y$.
Exponential — $f(x) = k^x$ The change in $y$ for a small step in $x$ is proportional to the current value of $y$ itself — the larger the function already is, the faster it grows from there. This is the conceptual seed of why exponential growth feels like it “runs away.”
Logarithmic — $f(x) = \log_m x$ The opposite behaviour: for the same small step in $x$, the change in $y$ gets smaller and smaller as $x$ grows larger. Early on, near small $x$, the function is very sensitive to a step; far out, the same step barely moves $y$ at all.
Trigonometric — $f(x) = \sin x$, $\cos x$ The rate of change is itself not constant but cyclical — fastest at some points, momentarily zero at others (where the curve flattens at a peak or a trough), and the pattern of speeding up and slowing down repeats forever.
The Five Primitive Functions, First-Principles Derivation
The derivative of a function is defined as
$$f'(x) = \lim_{\epsilon \to 0} \frac{f(x+\epsilon) – f(x)}{\epsilon}$$
Constant — $f(x) = k$
$$f'(x) = \lim_{\epsilon \to 0} \frac{k – k}{\epsilon} = \lim_{\epsilon \to 0} \frac{0}{\epsilon} = 0$$
Linear — $f(x) = ax + c$
$$f'(x) = \lim_{\epsilon \to 0} \frac{[a(x+\epsilon)+c] – [ax+c]}{\epsilon} = \lim_{\epsilon \to 0} \frac{a\epsilon}{\epsilon} = a$$
Power — $f(x) = x^n$
$$f'(x) = \lim_{\epsilon \to 0} \frac{(x+\epsilon)^n – x^n}{\epsilon}$$
Expanding $(x+\epsilon)^n$ by the binomial theorem,
$$(x+\epsilon)^n = x^n + nx^{n-1}\epsilon + \binom{n}{2}x^{n-2}\epsilon^2 + \cdots + \epsilon^n$$
$$f'(x) = \lim_{\epsilon \to 0} \frac{nx^{n-1}\epsilon + \binom{n}{2}x^{n-2}\epsilon^2 + \cdots + \epsilon^n}{\epsilon} = \lim_{\epsilon \to 0} \left[ nx^{n-1} + \binom{n}{2}x^{n-2}\epsilon + \cdots \right] = nx^{n-1}$$
Exponential — $f(x) = k^x$
$$f'(x) = \lim_{\epsilon \to 0} \frac{k^{x+\epsilon} – k^x}{\epsilon} = \lim_{\epsilon \to 0} \frac{k^x(k^\epsilon – 1)}{\epsilon} = k^x \lim_{\epsilon \to 0} \frac{k^\epsilon – 1}{\epsilon}$$
The remaining limit is a constant depending only on $k$, conventionally written $\ln k$, so
$$f'(x) = k^x \ln k$$
When $k = e$, $\ln e = 1$, giving the special case
$$f'(x) = e^x$$
Logarithmic — $f(x) = \ln x$
$$f'(x) = \lim_{\epsilon \to 0} \frac{\ln(x+\epsilon) – \ln x}{\epsilon} = \lim_{\epsilon \to 0} \frac{1}{\epsilon}\ln\left(\frac{x+\epsilon}{x}\right) = \lim_{\epsilon \to 0} \frac{1}{\epsilon}\ln\left(1+\frac{\epsilon}{x}\right)$$
Letting $h = \epsilon/x$, so $\epsilon = hx$ and $h \to 0$ as $\epsilon \to 0$,
$$f'(x) = \lim_{h \to 0} \frac{1}{hx}\ln(1+h) = \frac{1}{x}\lim_{h \to 0} \frac{\ln(1+h)}{h} = \frac{1}{x}$$
since
$$\lim_{h \to 0} \frac{\ln(1+h)}{h} = 1$$
Trigonometric — $f(x) = \sin x$
$$f'(x) = \lim_{\epsilon \to 0} \frac{\sin(x+\epsilon) – \sin x}{\epsilon}$$
Using $\sin(x+\epsilon) = \sin x \cos\epsilon + \cos x \sin\epsilon$,
$$f'(x) = \lim_{\epsilon \to 0} \frac{\sin x(\cos\epsilon – 1) + \cos x \sin\epsilon}{\epsilon} = \sin x \cdot \lim_{\epsilon \to 0}\frac{\cos\epsilon – 1}{\epsilon} + \cos x \cdot \lim_{\epsilon \to 0}\frac{\sin\epsilon}{\epsilon}$$
Using the two standard limits
$$\lim_{\epsilon \to 0}\frac{\cos\epsilon – 1}{\epsilon} = 0, \qquad \lim_{\epsilon \to 0}\frac{\sin\epsilon}{\epsilon} = 1$$
$$f'(x) = \sin x \cdot 0 + \cos x \cdot 1 = \cos x$$
Trigonometric — $f(x) = \cos x$
$$f'(x) = \lim_{\epsilon \to 0} \frac{\cos(x+\epsilon) – \cos x}{\epsilon}$$
Using $\cos(x+\epsilon) = \cos x \cos\epsilon – \sin x \sin\epsilon$,
$$f'(x) = \lim_{\epsilon \to 0} \frac{\cos x(\cos\epsilon – 1) – \sin x \sin\epsilon}{\epsilon} = \cos x \cdot \lim_{\epsilon \to 0}\frac{\cos\epsilon – 1}{\epsilon} – \sin x \cdot \lim_{\epsilon \to 0}\frac{\sin\epsilon}{\epsilon}$$
Using the two standard limits
$$\lim_{\epsilon \to 0}\frac{\cos\epsilon – 1}{\epsilon} = 0, \qquad \lim_{\epsilon \to 0}\frac{\sin\epsilon}{\epsilon} = 1$$
$$f'(x) = \cos x \cdot 0 – \sin x \cdot 1 = -\sin x$$
Final Formula List
- $$\frac{d}{dx}(k) = 0$$
- $$\frac{d}{dx}(ax+c) = a$$
- $$\frac{d}{dx}(x^n) = nx^{n-1}$$
- $$\frac{d}{dx}(k^x) = k^x \ln k, \qquad \frac{d}{dx}(e^x) = e^x$$
- $$\frac{d}{dx}(\ln x) = \frac{1}{x}$$
- $$\frac{d}{dx}(\sin x) = \cos x$$
- $$\frac{d}{dx}(\cos x) = -\sin x$$