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For positive measured inputs and $f=C{\prod }_i{x}_i^{a_i}$, logarithmic differentiation gives $df/f={\sum }_i{a}_{i}d{x}_i/x_i$. Applying propagation of uncertainty yields the displayed covariance formula. For independent input errors it reduces to ${\sigma }_f/|f|\simeq [{\sum }_i(a_i{\sigma }_i/x_i{)}^2{]}^{1/2}$. If the input uncertainties are strict bounds rather than standard deviations, a first-order conservative bound is $|\Delta f|/|f|\lesssim {\sum }_i|a_i|\Delta x_i/|x_i|$. These are different statistical interpretations; a product formula alone does not choose between them.

For real $x>0$, use logarithmic differentiation: $\log y=\tan x\log x$. The product rule gives $y^{\prime }/y={\sec }^{2}x\log x+\tan x/x$, henceWork on an interval where $\tan x$ is defined. The positive-base condition makes the variable real power unambiguous.

Use logarithmic differentiation on the positive function: $\ln f=x^{-1}\ln (x^2+2)$. The product rule and chain rule giveThe positive base permits this real definition on both components of the indicated domain. There is no need to treat a variable exponent as a constant.

For $x>0$, logarithmic differentiation givesSince $f(x)>0$, the derivative vanishes precisely when $\ln x=1$. HenceThe derivative is positive before $e$ and negative after $e$, so this is a global maximum on the given positive domain.

Converting all lengths to metres gives $d=3.7276\times 10^{-5}\mathrm{m}$. Logarithmic differentiation givesIf the quoted uncertainties are independent standard uncertainties, monomial uncertainty propagation givesIf the quoted uncertainties instead mean bounds, the first-order conservative fractional error is $0.111$, giving approximately $37.3\pm 4.1\mu \mathrm{m}$. The question does not specify their statistical meaning; these two error conventions should not be combined.

For the same massless spring of spring constant $k$, Hooke's law gives the vertical simple harmonic motion about the displaced equilibrium position:Gravity shifts the equilibrium extension but does not change this frequency. Use the measured sample means in the ratio estimate. For small measurement errors, logarithmic differentiation givesAssuming the two frequency estimates have independent errors, their variances add. Writing $ϵ=\delta \nu /\overline{\nu }$ for their common fractional standard uncertainty, monomial uncertainty propagation yieldsThis is a first-order uncertainty formula, rather than an exact identity for an arbitrary noisy ratio. It assumes that errors in the two measured frequencies have zero covariance.

There is a statistical distinction in the wording: the standard deviation of individual repetitions is not the standard error of their sample mean. If the supplied $\sigma ({\nu }_i)$ describes individual trials, the frequency means have errors $\sigma ({\nu }_i)/\sqrt{n}$, and the ratio formed from those means instead satisfiesThus the printed formula describes either a single-pair ratio or a convention in which $\delta \nu$ already denotes the uncertainty of each averaged frequency. Under either consistent interpretation, independent repeated measurements reduce statistical error in proportion to $n^{-1/2}$. Halving it requires $4n$ trials in total for each mass, or $3n$ additional trials. Repetition alone does not reduce a fixed systematic error.

Rearrange the supplied form of Kepler's third law to make the gravitational constant the subject:Convert the semi-major axis and orbital period to SI units before substitution:The central value is thereforeFor monomial uncertainty propagation, logarithmic differentiation givesIf the quoted errors are independent standard uncertainties, the variances of these contributions add:Thus ${\sigma }_G=1.61335\times 10^{-12}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and the measured result under that statistical convention isThe semi-major axis measurement supplies the largest individual contribution to the relative uncertainty.

The paper does not specify whether its errors are statistical uncertainties or bounds. If they are maximum input errors, the first-order worst-case error propagation estimate adds magnitudes:Correlation between input errors would instead require the covariance terms in the general propagation of uncertainty formula; they are not specified.

The vibroscope area estimator converts the fundamental frequency of a tensioned material fibre into a cross-sectional area through a mechanical model. For a uniform flexible strand of length $L$, positive tension $T$, linear mass density $\mu$ and mass density $\rho$, the fixed-end string relation givesThe second equation uses $\mu =\rho A$: the frequency first identifies mass per length under the stated tension, and a chosen mass density then converts it into area. Circular geometry is needed only to infer a diameter from that area.^[ref]

The experimental study by Bo Madsen, Rajnish Kumar and Hans Lilholt used a Favimat+ fibre testing machine. Acoustic waves excited the strand, and a frequency sweep plus a polynomial fit located the resonance peak. A load cell measured tension, while grip separation supplied $L$. The reported frequency range was 500–4000 Hz, with an estimated resolution of 0.1 Hz; the stated force and length readout resolutions were $1\mu \mathrm{N}$ and $0.1\mu \mathrm{m}$. Those readout resolutions do not themselves specify the estimator's full measurement uncertainty.^[ref]

The area and mass density must refer to the same loaded or reference configuration. The companion theory derives volume and area corrections for recovering an untensioned reference area from the tensioned measurement.^[ref]

The vibroscope area uncertainty follows locally by logarithmic differentiation:Treat the small input errors as independent random variables, and let each $u$ denote the corresponding standard uncertainty. First-order uncertainty propagation then givesCorrelated inputs require their covariances as well. Four independent 1% relative standard uncertainties give $\sqrt{10}\%\approx 3.16\%$ for the area. A systematic 1% overestimate of frequency alone instead multiplies the area estimate by $(1.01{)}^{-2}\approx 0.9803$: it produces about a 1.97% underestimate.

Bending resistance can also increase the observed frequency. The companion theory paper models fully clamped supports and derives the corresponding frequency equation and deformation corrections.^[ref] An independently derived, simply supported example isolates the bending-stiffness bias in the vibroscope area estimator. Let $B=EI$ be the filament bending modulus and impose zero deflection and zero bending moment at each endpoint. Substituting $y=\sin (n\pi x/L)\cos (2\pi f_{n}t)$ into the tensioned Euler-Bernoulli beam equation $\mu y_{tt}+B{y}_{xxxx}-T{y}_{xx}=0$ givesWith $r=b/a={\pi }^{2}B/(T{L}^2)$, the uncorrected flexible-string estimate satisfies $\hat{A}/A=1/(1+r)$. At $r=0.10$, it is 9.09% too small. The same model yields two-mode separation of tension and bending stiffness:For $T>0$ and $B\ge 0$, the predicted mode ratio is $f_2/f_1=2\sqrt{(1+4r)/(1+r)}$, ranging from 2 toward 4. These identities belong to the simply supported model; clamped grips require their own boundary conditions. Neither a single resonance peak nor an assumed diameter establishes the appropriate support, density or bending correction.