It occurs to me that maybe you can approach this from the viewpoint of approximations. Briefly, I’m thinking of the kinds of approximations they likely use in practice, and which are often given in an appendix at the back of texts, such as the following:
For $x \approx 0$ we have $\frac{1}{1-x} \approx 1+x$ (multiply numerator and denominator of left side by $1+x$ and then drop the $x^2$ term).
For $\theta \approx 0$ we have $\sin \theta \approx \theta$ and $\tan \theta \approx \theta$ (use a geometric argument).
For $\delta \approx 0$ we have $\sqrt{a^2 + \delta} \; \approx \; a + \frac{\delta}{2a}$ (see here for an algebraic derivation).
Both the product rule for derivatives (expand $(u + \Delta u)(v + \Delta v)$ and drop the $“\text{small} \times \text{small}”$ term) and the quotient rule for derivatives (multiply numerator and denominator of $\frac{u + \Delta u}{v + \Delta v}$ by $v - \Delta v$ and drop the two $“\text{small} \times \text{small}”$ terms) can be heuristically obtained by methods that they will be (or at least should be) familiar with. The idea is that you want the simplest non-constant approximation, which winds up being a linear (first degree) approximation, and this arises as the linear part when you are able to expand the quantity as a (or some simple algebraic combination of) possibly infinite sum(s) of non-negative integer powers of “small things”. Then maybe briefly discuss aspects of (a) why we can drop the higher order small terms (this graphical exploration idea is worth considering) and (b) the value of a linear approximation (easy to integrate linear functions, can apply proportionality when linear, simple to numerically calculate or estimate when linear, etc. --- note that I’m trying to avoid simply saying “linear” is good or “graph locally looks like a line” is good, because this leaves unanswered why linear/lines are good/useful in the first place).
Finally, the books I list in my answer to Teaching Calculus Less Formally (especially Calculus Made Easy by Silvanus P. Thompson and Calculus in Context. The Five College Calculus Project by James Callahan, et al.) might be worth looking through for ideas in general.