In Chapter 2, section 2.5.1, Boyd and Vandenberghe give a proof of the Separating Hyperplane theorem. They base their proof on two points, $c \in \mathcal{C}$ and $d \in \mathcal{D}$, that are the pair of points in the two convex sets that are the closest to each other.
The hyperplane that separates the sets is expressed as $a^\top x = b$. Then, they define $$a = d - c$$ and $$ b = \frac{||d||_2^2 - ||c||_2^2}{2}$$
Then they set up to show that the function $f(x) = a^\top x - b = (d - c)^\top(x - (1/2)(d-c))$ is nonpositive on C and nonegative on D. They show that $f$ is nonnegative on D the following way:
Suppose there were a point $u \in \mathcal{D}$ for which $f(u) = (d - c)^\top(u - (1/2)(d-c)) <0$. We can express $f(u)$ as $f(u) = (d - c)^\top (u - d + (1/2)(d-c)) = (d-c)^\top (u-d) + (1/2)||d-c||_2^2$. That implies $(d-c)^\top (u-d) < 0$.
Then, they make the following statement: "We observe that $$ \frac{d}{dt} ||d + t(u-d) - c||_2^2\Bigr|_{\substack{t=0}} = 2(d-c)^\top (u-d) <0 $$
Up until the last equation I followed the proof with no issue. But I can't see the reason they had to introduce a derivative on $t$. Is that something natural to the proof or was that an artifice to help them show their point (which I missed)?