Here is a very general but broad class of applications: Suppose you have some quantity $q(t)$ that you want to model with respect to time, like maybe a population, or a chemical concentration, or an object's speed, or whatever. Quite often there will be a natural way to describe the quantity you're interested in by using a differential equation, i.e. an equation which relates the rate of change $\frac{dq}{dt}$ of the quantity to $q(t)$ itself.
Calculus can then be used to analyze the differential equation (which could be very complicated) and hopefully give a closed-form solution so that we can predict the quantity in the long term. If an explicit solution is found, calculus can again be used to analyze the solution to find maxima and minima, and all sorts of critical points of interest.
Differential equations aren't only useful for modelling quantities, but also positions. for example, in order to fully understand how a rocket ship blasts off into space, scientists need to take into account the fact that the burning of fuel means the mass is decreasing, and so the propulsion will cause a larger acceleration. This problem leads to solving a differential equation.
At my undergraduate institution (Facultad de Ciencias, UNAM, Mexico), for a while in the mid-to-late 70s, several professors in the Calculus sequence (four courses: Differential single-variable (Calc I), Integral single-variable (Calc II), Differential multi-variable (Calc III), Integral multi-variable (Calc IV)) decided to use Hasse's analysis textbook instead of a calculus textbook. It was more of a "baby analysis" than a calculus course.
Now, this was done only in courses that were being taught to Math, Actuarial Sciences, and Physics majors (and a Math major takes nothing but math courses, for instance).
It did not go well. Students didn't learn analysis very well, and they certainly did not learn the calculus skills they needed very well. The Physics department, in particular, went up in arms because the Physics majors were coming out of these courses unable to actually compute integrals and derivatives, or use them to solve specific physics problems. Same problem with the actuarial scientists. The math majors fared a little better, but mainly because the same people who were doing this were the people who were also teaching the analysis courses in the junior and senior years; but those that went on to take analysis from other people didn't do so well. In addition, the failure rate for these courses was extremely high. (Failure rate in the Calculus sequence has always been way too high there, but it got much worse).
Most professors switched back to calculus books and to not do baby analysis. By the mid-80s, almost nobody was using Hasse's book or teaching "mini-analysis."
If a student has had a good enough calculus course in High School, then it is likely that a baby analysis course might indeed be beneficial, building on the bases that calculus can help set. This could very well be the case in the EU; it's not the case in the US. (In Mexico, nominally, students in the Math/Physics/Engineering track were taking a year of Calculus as seniors in High School, but obviously not good enough).
Best Answer
In terms of physics, we have velocity and acceleration.
In terms of economics, derivative appears in elasticity and it helps you to understand the impact of adjusting the price on your revenue.
For a function that is differentiable, derivative help you to find the optimal value. Optimization is used widely in machine learning. For example, gradient descent based algorithm are used in learning the parameters of neural network to help you make good predictions.