Differentials

A differential is simply a small change, and $dy = f'(x)\,dx$ says how a small change in the input becomes a change in the output. All we're really saying is: if $y$ changes 3 times as fast as $x$, then nudging $x$ by $0.1$ moves $y$ by about $0.3$. The pieces are $dx$ (the small nudge to the input), $f'(x)$ (how sensitive the output is right there) and $dy$ (the resulting change, measured along the tangent line). The true change, $\Delta y$, follows the curve instead, but for small nudges the two are nearly the same.

If that sounds familiar, it should. Differentials are last lesson's linear approximation written as changes. $L(x) = f(a) + f'(a)(x - a)$ says the new value is the old value plus $f'(a)$ times the step. Move $f(a)$ to the other side and call the step $dx$, and what's left is $dy = f'(a)\,dx$: instead of approximating $f(x)$, we're approximating how much $f$ changes.

Error propagation. If a measurement $x$ has error $dx$, a quantity $y = f(x)$ computed from it has error about $dy = f'(x)\,dx$. Where $f'$ is large, the formula magnifies your measurement error; where it's small, it shrinks it.

Relative error is $\frac{dy}{y}$. For powers it has a neat rule: if $y = x^n$, then $dy = nx^{n-1}\,dx$, and dividing by $y = x^n$ gives $\frac{dy}{y} = n\frac{dx}{x}$. A 1% error in a radius becomes about a 3% error in a sphere's volume, because volume goes like $r^3$.

Every lab report's error bars come from differentials. Engineers use them to decide which measurement needs a better instrument: the one whose errors get magnified most.