Differential Calculus

Any physical quantity that changes with another (usually time) is represented mathematically by a function. Temperature through the day is written $T(t)$ — 'temperature as a function of time' — meaning that once a particular time $t$ is given, the function $T(t)$ returns the temperature at that instant. In exactly the same way, the position of a particle moving along the $x$-direction is written $x(t)$, 'x as a function of time'.

More generally, if $y=f(x)$, then $y$ is called the dependent variable and $x$ the independent variable: as $x$ changes, $y$ changes with it. Calculus is the branch of mathematics built to analyse exactly how a quantity changes as its independent variable changes, and the derivative $dy/dx$ is its central tool — it measures the instantaneous rate of change of $y$ with respect to $x$. Formally,

Graphically, $\Delta y/\Delta x$ is the slope of the chord joining two nearby points on the curve $y(x)$; as $\Delta x$ is shrunk towards (but never quite reaches) zero, that chord settles down to the tangent line at the point $x$, and $dy/dx$ is the slope of that tangent. (It is important that $\Delta x\to 0$ does not mean $\Delta x=0$ — the ratio $\Delta y/\Delta x$ is undefined at $\Delta x=0$, but it approaches a perfectly well-defined limiting value as $\Delta x$ gets arbitrarily small.)

A short table of derivatives used constantly in this chapter and beyond: