Great minds think alike! Highfive Mike! :D

Wow, associated professor of Mathematics! Respect!

Are you the gigachad that teaches Multivariable Calculus here at Wolfram U? That's where I'm heading next after this course!

Tingting,

While @Luke Titus's math is impeccable, as usual, I'm with you about "replace" sounding weird. Consider ${d \over dx} \cos(y)$, where $y$ is to be considered an implicit function of $x$. Now let's treat $\cos(y)$ as its own expression and replace $y$ by $dy/dx$. What do we get?:

$$ a)\ \ \cos\left({dy \over dx}\right) \quad b)\ \ {-}\sin\left({dy \over dx}\right) \quad c)\ \ {-}\sin(y) \cdot {dy \over dx}$$

Both a) and b) replace $y$ by $dy/dx$. Of course, c) is the correct answer, but $dy/dx$ is not replacing anything. A new factor is being inserted (as is required by the Chain Rule).

So, yes, I think they should make a note to consider revising this description for the next edition.