# Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?

This answer, to a question on teaching the chain rule, suggests writing something like this $$\frac{\mathrm{d}\, \mathrm{e}^\sqrt{s}}{\mathrm{d}\,s}=\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\sqrt{s}}\frac{\mathrm{d}\sqrt{s}}{\mathrm{d}\,s}$$ where a composite term ($\sqrt{s}$) is substituted for the variable occurring in the denominator of a derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$.

It is still common today to substitute a term for $y$ in $\frac{\mathrm{d}y}{\mathrm{d}x}$, but I have rarely (or never) seen people substitute for $x$ in $\frac{\mathrm{d}y}{\mathrm{d}x}$. On the other hand it makes a lot of sense, when the "differential coefficient" $\frac{\mathrm{d}y}{\mathrm{d}x}$ is literally understood as "the coefficient in front of the differential $\mathrm{d}x$ in $\mathrm{d}y=c\mathrm{d}x$".

Question: Can one find expressions like $\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\sqrt{s}}$ in classical works of differential calculus? Was it popular at some time?

(The motivation for this question comes from another question I asked on formalizing classical calculus: https://cs.stackexchange.com/questions/82230)

• I'd be interested in understanding the reason for the downvote. Oct 20, 2017 at 13:34
• It's conceivable that someone thought that substitution or change of variables in solving differential or integral equations might be too common to be remarkable here? (btw I didn't have any part in making that vote). Oct 21, 2017 at 11:32
• @terry-s that‘s conceivable. Yet, although substitution is common, I have not seen people write derivatives like that. Oct 21, 2017 at 11:45
• In answer to the question you may find some interesting stuff in an area known as Fractional Calculus. Oct 22, 2017 at 21:00

Gibbs (1889, p. 140): $$\qquad \dfrac{d\,\log\mathrm V}{d\,\log p} = - \dfrac{d\,\log n}{d\,\log\lambda}$$

Riemann (1868, p. 89): $$\qquad \dfrac{d^2y}{dx^2}-\dfrac1{\alpha\alpha}\dfrac{d^2y}{dt^2}=4\dfrac{d\smash[t]{\dfrac{dy}{d(x+\alpha t)}}}{d(x-\alpha t)}$$

Hamilton (1843, p. 362): $$\quad\ \dfrac d{d\log t}\,(1+t)^{-m}\,\mathrm T_m$$

Jacobi (1841, p. 336): $$\qquad \dfrac{\partial R}{\partial a}= \sum\cdot \dfrac{\partial R}{\partial \frac{\partial f_i}{\partial x_k}}\cdot\dfrac{\partial^2 f_i}{\partial a\,\partial x_k}$$

Legendre (1826, p. 466): $$\quad \dfrac{d\,l\,\Gamma a}{da} + \dfrac{d\,l\,\Gamma(\frac12+ a)}{d(\frac12+ a)} - \dfrac{2d\,l\,\Gamma(2a)}{d(2a)} = -2l\,2$$

Gauss (1809, p. 27): $$\qquad \dfrac{\mathrm{d}\frac12v}{\mathrm{d}\log\operatorname{tang}\frac12v}= \dfrac{\sin v}2$$

L’Huilier (1795, p. 96): $$\qquad \dfrac{d.a^{\mathrm z}}{d.\log.z}= A.a^{\mathrm z}$$