More specifically, was Newton aware that given an inverse pair of functions $f$ and $h$ such that
$$f(h(x)) = x = h(f(x))$$ about the origin that, for
$$(x,y)=(h(y),f(x)),$$
the derivatives satisfy
$$f^{'}(x) = 1/h^{'}(y)$$
or
$$dy/dx = 1/(dx/dy)$$
near the origin?
Heuristically, this follows symbolically from
$$dy = f^{'}(x)dx = f^{'}(x)h^{'}(y)dy, $$ or, equivalently, from the chain rule applied to the top equation.
And it follows geometrically for a function whose graph lies in the first quadrant by reflection through the bisector of the first quadrant, the line $y=x$. Clearly, the slope for any tangent line is inverted by the reflection just as displacements along the $x-$axis and the $y-$axis are interchanged. In fact, it follows directly from the tangent line perspective since $$ y = m \; x + b$$ and $$y = \frac{1}{m}(x-b)$$ describe an inverse pair.
Surely, with Newton's mastery of geometric calculus, he was aware of these relationships. Is there evidence of this in Newton's work?
Related MO-Q by Ziegler.
Cross-posted from this MO-Q.
Edit 6/12/17:
An example of a calculation incorporating the IFT that would have been obvious to Newton and plausible for him to have performed if only as a simple check of his general formulas:
It was known well before Newton that
$$\frac{d\tan(x)}{dx} = 1+ \tan^2(x),$$
or, with $y = \tan(x)$,
$$\frac{dy}{dx} = 1+ y^2.$$
In terms of fluxions and fluents, this could be put in the form of Newton's implicit function
$$g(x,y,\dot{x},\dot{y})=\dot{y}-(1+y^2)\dot{x}=0.$$
Then
$$\frac{\dot{x}}{\dot{y}}= \frac{1}{1+y^2}=\frac{dx}{dy}, $$
and application of the binomial theorem and integration would give the series
$$ \arctan(y) = x = y - \frac{y^3}3+\frac{y^5}5-\frac{y^7}7+\dots. \tag3 $$
Newton could then have derived a series expression for $\tan(x)$ using his series reversion formula (see Ferraro) for finding the series for the compositional inverse of a function from its power series. In fact, the same procedure is applied to finding a series for $\sin(x)$ in Ferraro on pages 76-78 following an alleged reconstruction by Horsley of Newton's derivation of the series.
Edit (Apr 10, 2018):
According to the Wikipedia article on the chain rule, both Newton and Leibniz were aware of the chain rule, and the inverse function theorem in its simplest form follows from application of the chain rule to $x = f(f^{-1}(x))$. This would provide an easy check for the veracity of the chain rule that someone as fastidious as Newton would have used.
In a certain sense it is obvious that Newton knew this at least after 1684, since Leibniz published his foundational article on infinitesimal calculus that year. In that article Leibniz introduces the notation $\frac{dy}{dx}$ (the quotient of infinitesimal differentials) and from this point of view your observation is obvious. The mathematicians in England may have been late in adopting Leibnizian notation but Newton was surely aware of the scientific developments on the continent.