Quick doubt on an application of the chain rule

Question

In Perloff's Microeconomics With Calculus 3rd edition, on page 63, one proceeds to the differentiation of the following equation with respect to $\tau$: $$D(p(\tau))=S(p(\tau)-\tau)$$ And the result is: $$\frac{\text{d}D}{\text{d}p}\frac{\text{dp}}{\text{d}\tau}=\frac{\text{d}S}{\text{d}p}\frac{\text{d}(p(\tau)-\tau)}{\text{d}\tau}=\frac{\text{d}S}{\text{d}p}(\frac{\text{d}p}{\text{d}\tau}-1)$$ My doubt is: why is the third member of the equation equal to the one in the middle? I really can't see how to go from one to another. Besides, according to the chain rule, shouldn't the derivative of $S(p(\tau)-\tau)$ with respect to $\tau$ be $\frac{\text{d}S}{\text{d}(p(\tau)-\tau)}\frac{\text{d}(p(\tau)-\tau)}{\text{d}\tau}$? Any help will be much appreciated. Thanks very much in advance.

Answer 1

It appears to be correct. .. Apparently $p= p (\tau )-\tau $, by perhaps slight abuse of notation. ..

For the second equality note that $\frac {d\tau}{d\tau }=1$ and use linearity of the derivative. ..