Economics has a ton of applications since they are interested in relationships between variables, not functions per se. Two simplish ones that come to mind are
Price elasticity of demand: Supply curves are just a relation between supply and demand, it need not be functional. You can derive the price elasticity in this setting. Here's are some good notes from Y. Katznelsons course at UCSC. A typical problem looks like: Price $p$ and quantity $q$ are related by $p^2 - pq + q^2 = 400$. When the price is \$4, estimate the percent change in profit when the price is raised 1%.
To generalize the above, comparative statics uses implicit differentiation to study the effect of variable changes in economic models. Here's a decent here's a decent introduction
Preference bundles, utility and choice. You have to gloss over some machinery but you're essentially doing calculus on level curves. If this looks interesting peek into the first few chapters of Hal Varian's Intermediate Microeconomics.