the delta method
Suppose you know roughly the mean and variance of some random quantity X, and now you care about a function of it, g(X). Recomputing the exact distribution of g(X) might be hard or impossible. The delta method gives a quick, approximate answer for the mean and variance of g(X) using nothing but calculus — the first derivative of g.
The idea is to replace the curve g by its tangent line near the typical value of X. If X clusters around its mean mu, then near mu we have g(X) approximately equal to g(mu) + g'(mu) (X - mu) — a straight-line approximation. A linear function shifts and scales, so reading off the mean and variance is immediate: E[g(X)] is approximately g(mu), and Var(g(X)) is approximately (g'(mu))^2 times Var(X). The slope g'(mu) tells you how strongly the function magnifies or damps the input's wobble: a steep g amplifies the variance, a flat g shrinks it. That single squared-slope factor is the heart of the method.
This is the standard engine of error propagation in science and the workhorse behind asymptotic standard errors in statistics: take an estimator that is roughly normal around the true value, push it through g, and the delta method hands you the approximate standard error of g of the estimator. Honest limits: it is a first-order (linear) approximation, accurate only when the variability of X is small relative to the curvature of g, and it can mislead near a point where g'(mu) = 0 (there the linear term vanishes and you need the next, second-order term).
Let X have mean 100 and standard deviation 5, and let g(X) = ln(X). Then g'(x) = 1/x, so g'(100) = 0.01. The delta method gives E[ln X] approximately ln(100) and Var(ln X) approximately (0.01)^2 times 25 = 0.0025, i.e. a standard deviation of about 0.05 on the log scale. The small slope 0.01 has shrunk a spread of 5 down to about 0.05.
Replace g by its tangent line at the mean; the squared slope scales the variance.
It is a first-order approximation: reliable only when X's spread is small relative to how much g curves. Near a point where g'(mu) = 0 the linear term vanishes, so a second-order delta method is needed instead.