Quotient Rule

The quotient rule differentiates a ratio of two functions. Ratios are everywhere in epidemiology — prevalence proportions, hazard ratios, and the logistic curve are all quotients whose rates of change we often need. Frequency-dependent transmission βSI/N\beta S I / N and prevalence I/NI/N are ratios of quantities that all change over time, so differentiating such a per-capita rate calls for the quotient rule.

The rule

ddx[f(x)g(x)]=f′(x) g(x)−f(x) g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{\big[g(x)\big]^2}

Order matters in the numerator: it is “derivative of top times bottom, minus top times derivative of bottom,” all over the bottom squared.

Intuition

Write the quotient as a product f⋅g−1f \cdot g^{-1} and apply the product and chain rules:

ddx[fg−1]=f′g−1+f (−g−2g′)=f′g−fg′g2=f′g−fg′g2.\frac{d}{dx}\big[f g^{-1}\big] = f' g^{-1} + f\,(-g^{-2} g') = \frac{f'}{g} - \frac{f g'}{g^2} = \frac{f' g - f g'}{g^2}.

So the quotient rule is not a new idea — it is the product rule in disguise.

Worked example: the logistic-type ratio x1+x\dfrac{x}{1+x}

Let f(x)=xf(x) = x and g(x)=1+xg(x) = 1 + x, so f′(x)=1f'(x) = 1 and g′(x)=1g'(x) = 1:

ddx[x1+x]=(1)(1+x)−x(1)(1+x)2=1(1+x)2.\frac{d}{dx}\left[\frac{x}{1+x}\right] = \frac{(1)(1+x) - x(1)}{(1+x)^2} = \frac{1}{(1+x)^2}.

The derivative is always positive, so x1+x\frac{x}{1+x} increases toward its saturating limit of 11 — the same “diminishing returns” shape as a saturating incidence or dose-response curve. At x=1x = 1 the slope is 14\tfrac{1}{4}.

Computing it

R

# Symbolic
D(expression(x / (1 + x)), "x")
#   1/(1 + x) - x/(1 + x)^2   == 1/(1+x)^2

# Numeric check at x = 1
library(numDeriv)
grad(function(x) x / (1 + x), 1)   # 0.25

Python

import sympy as sp
x = sp.symbols("x")
sp.simplify(sp.diff(x / (1 + x), x))   # 1/(x + 1)**2

# Numeric check at x = 1
h = 1e-6
f = lambda x: x / (1 + x)
(f(1 + h) - f(1 - h)) / (2 * h)        # ~0.25

Julia

using Symbolics
@variables x
simplify(Symbolics.derivative(x / (1 + x), x))   # (1 + x)^-2

using ForwardDiff
ForwardDiff.derivative(x -> x / (1 + x), 1.0)     # 0.25

Why it matters for statistics

Proportions, rates, and probabilities are ratios, and models like logistic regression are built from them. Differentiating these quotients is how we obtain the score functions and delta-method standard errors for estimated proportions and odds.