= Solution
Use <stellar homology>: dimensionless density, <temperature> and <luminosity> profiles are taken to have the same shape, so the dimensional scale relations follow from the structure equations. <Mass conservation> and hydrostatic support give
$$
\rho_c\propto\frac M{R^3},\qquad P_c\propto\frac{GM^2}{R^4}.
$$
The <ideal gas> equation then gives $T_c\propto\mu M/R$, with the dimensional constants held fixed. The specific <stellar energy-generation rate> has the prescribed <hydrogen> factor and <temperature> exponent, so
$$
L_{\mathrm{nuc}}\propto\epsilon_cM
\propto\epsilon_0X\rho_cT_c^{13}M
\propto\epsilon_0X\mu^{13}M^{15}R^{-16}.
$$
Independently, <radiative diffusion in a star> gives $T_c/R\propto\kappa\rho_cL/(R^2T_c^3)$, hence
$$
L_{\mathrm{rad}}\propto\frac{RT_c^4}{\kappa\rho_c}
\propto\frac{\mu^4M^3}{\kappa}.
$$
Equating the nuclear supply and radiative transport establishes <composition-dependent CNO homology>:
$$
\boxed{R\propto[X\kappa\mu^9]^{1/16}M^{3/4},\qquad
L\propto\mu^4\kappa^{-1}M^3.}
$$
At one fixed composition, these reduce to $R\propto M^{3/4}$ and $L\propto M^3$ as required.
The <Stefan–Boltzmann law> gives $T_e^4\propto L/R^2\propto M^{3/2}$, so $T_e\propto M^{3/8}$. Thus the logarithmic <Hertzsprung-Russell diagram> slope along this <zero-age main sequence> is
$$
\boxed{\frac{d\log L}{d\log T_e}=8.}
$$
Increasing <mass> moves upward and leftward on the conventional diagram, whose <temperature> axis is reversed.
For composition changes, neglect the small heavy-element contribution to the <mean molecular weight> and set $Y\simeq1-X$. Then $\mu^{-1}=(3+5X)/4$. The fixed trace catalyst abundance is absorbed into $\epsilon_0$; this is not an assertion that an exactly metal-free star can undergo CNO burning. Substitution into the homology laws gives
$$
\boxed{L\propto M^3(1+X)^{-1}(3+5X)^{-4},}
$$
$$
\boxed{R\propto M^{3/4}X^{1/16}(1+X)^{1/16}(3+5X)^{-9/16}.}
$$
For a retained, nonnegligible fixed metallicity $Z$, the corresponding mean-molecular-weight factor is $3+5X-Z$ instead. The displayed requested formula uses the trace-metal approximation.
At fixed <mass>, differentiating with respect to $X$ at the formal reference $X=1$ gives
$$
\frac{d\log L}{dX}=-\frac1{1+X}-\frac{20}{3+5X}\ \longrightarrow\ -3,
$$
$$
\frac{d\log R}{dX}=\frac1{16X}+\frac1{16(1+X)}-\frac{45}{16(3+5X)}
\ \longrightarrow\ -\frac{33}{128}.
$$
Consequently
$$
\frac{d\log T_e}{dX}=
\frac14\left(\frac{d\log L}{dX}-2\frac{d\log R}{dX}\right)
\ \longrightarrow\ -\frac{159}{256},
$$
and the initial <fully mixed hydrogen-burning evolutionary track> has slope
$$
\boxed{\left.\frac{d\log L}{d\log T_e}\right|_{M,\,X=1}
=\frac{256}{53}\simeq4.83.}
$$
<Hydrogen> depletion has $dX<0$, so both <luminosity> and <effective temperature> initially increase. The star therefore moves upward and leftward, on a shallower logarithmic track than the <mass> sequence. The figure uses dimensionless <luminosity> and <temperature> relative to its initial reference star; the track is computed for declining $X$ at constant <mass>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-63-homology-track.png]
{title=Zero-age radiative homology sequence and the fixed-mass fully mixed hydrogen-burning track}
Ordinary cluster main-sequence turn-offs instead evolve toward brighter, cooler, more extended configurations and eventually the giant region. Normal stars are therefore not mixed throughout their entire <mass> during <hydrogen burning>: the core composition changes while the envelope retains <hydrogen>, and composition and entropy gradients develop. Core exhaustion and subsequent shell burning then produce very different structural evolution. Complete mixing is a special evolutionary scenario, not the standard explanation of ordinary giant formation.
These hydrogen-burning scalings must not be extrapolated literally to $X=0$. Their vanishing nuclear coefficient would make the inferred radius tend to zero and the <effective temperature> singular; in reality <hydrogen> exhaustion requires a different structural and energy-generation model. A homogeneous remnant may be helium-rich, but its later structure is not described by the same CNO law. Also, the cover-sheet radiation-pressure expression has a temperature-power typo: the physical term is $aT^4/3$. <Radiation pressure> was neglected in this derivation, so that typo does not enter the calculation.
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